ON THE STRONG LAW OF LARGE NUMBERS FOR WEIGHTEDSUMS OF RANDOM ELEMENTS IN BANACH SPACES

By

YUAN LIAO

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2012

c⃝ 2012 YUAN LIAO

2

To my teachers with gratitude

To my parents with love

3

ACKNOWLEDGMENTS

First and most importantly, I must convey my sincerest gratitude to my Ph.D

advisor, Professor Andrew Rosalsky, for his invaluable guidance and constant support

throughout my graduate studies. This dissertation would not have been possible without

his step-by-step guidance. He always generously shares his ideas and makes great

effort to explain them in the clearest way possible. I feel very fortunate to get to know

him. He is an amiable mentor full of enthusiasm for probability theory. He is always

patient to correct the faults in my work and provide the most informative and inspiring

feedback. I must admit that I have learned a lot from his meticulous academic attitude.

Next, I would like to thank everyone else on my supervisory committee: Distinguished

Professor Malay Ghosh, Dr. Kshitij Khare, and Dr. Amy Cantrell. I am grateful for all of

their support and help.

Finally, I would extend my earnest thanks to my parents, who have always been

confident and pride of me and encouraging me to chase my dreams. They have always

and forever been my inspiration !

4

TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.1 Strong Law of Large Numbers for Random Variables . . . . . . . . . . . . 81.2 Strong Law of Large Numbers for Banach Space Valued Random Elements 111.3 Motivation and Organization of Dissertation . . . . . . . . . . . . . . . . . 14

2 PRELIMINARIES: DEFINITIONS, LEMMAS, AND NOTATION . . . . . . . . . . 22

2.1 Basic Concepts of Banach Spaces . . . . . . . . . . . . . . . . . . . . . . 222.2 Probability in Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 252.3 Useful Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3 STRONG LAWS OF LARGE NUMBERS IN RADEMACHER TYPE p (1 ≤ p ≤ 2)BANACH SPACES FOR INDEPENDENT SUMMANDS . . . . . . . . . . . . . 42

3.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4 STRONG LAWS OF LARGE NUMBERS FOR RANDOM ELEMENTS IN GENERALBANACH SPACES IRRESPECTIVE OF THEIR JOINT DISTRIBUTIONS . . . 68

4.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5 FUTURE RESEARCH AND CONCLUSIONS . . . . . . . . . . . . . . . . . . . 91

5.1 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5

LIST OF FIGURES

Figure page

2-1 Expected Value of a Random Element in Lp(R), 1 ≤ p < ∞ . . . . . . . . . . . 32

6

Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

ON THE STRONG LAW OF LARGE NUMBERS FOR WEIGHTEDSUMS OF RANDOM ELEMENTS IN BANACH SPACES

By

YUAN LIAO

May 2012

Chair: Andrew RosalskyMajor: Statistics

Let Vn, n ≥ 1 be a sequence of random elements in a real separable Banach

space and suppose that Vn, n ≥ 1 is stochastically dominated by a random element

V . Let an, n ≥ 1 and bn, n ≥ 1 be real sequences with 0 < bn ↑ ∞. The main results

are strong laws of large numbers (SLLNs) obtained for the following two broad cases;

the results are new even when the underlying Banach space is the real line.

(i) Conditions are provided under which an(Vn − EVn), n ≥ 1 obeys a general

SLLN of the form∑ni=1 ai(Vi − EVi)/bn → 0 almost certainly where the Vn, n ≥ 1

are independent. The underlying Banach space is assumed to satisfy the geometric

condition that it is of Rademacher type p (1 ≤ p ≤ 2). Special cases include results

of Woyczynski (1980), Teicher (1985), Adler, Rosalsky, and Taylor (1989), and Sung

(1997).

(ii) Conditions are provided under which anVn, n ≥ 1 obeys a general SLLN of

the form∑ni=1 aiVi/bn → 0 almost certainly irrespective of the joint distributions of the

Vn, n ≥ 1. No geometric conditions are imposed on the underlying Banach space.

The results are general enough to include as special cases results of Petrov (1973),

Teicher (1985), Sung (1997), and Rosalsky and Stoica (2010).

Numerous examples are provided which illustrate, compare, or demonstrate the

sharpness of the results.

7

CHAPTER 1INTRODUCTION

1.1 Strong Law of Large Numbers for Random Variables

Probability theory, as a mathematical discipline concerned with the analysis of

random or chance phenomena, has developed not only profoundly in its all classical

branches but also widely from problems arising from other branches of science such as

mathematical statistics and physics. The essential components of probability theory are

experimental outcomes (called sample points), events, random variables, and stochastic

processes. The latter two are mathematical abstractions of non-deterministic measured

quantities that may either be a single value or evolve over time in a random fashion.

If a random experiment is repeated many times, a sequence of random events will

demonstrate certain patterns which can be studied and predicted. Two representative

mathematical results describing such patterns are the law of large numbers and the

central limit theorem, which have been crowned as being the first two of the three

pearls of probability theory. (The law of iterated logarithm, the third pearl of probability

theory, has not yet had as big an impact on applications that can be compared with

the other two, since it cannot be observed even in a large number of replications of the

experiment.)

The laws of large numbers have become the stepping stone between probability

theory and mathematical statistics. On one hand, the basic goal of probability theory is

to calculate the probabilities of events under a given probabilistic model. On the other

hand, mathematical statistics in a certain sense handles the inverse of the problems

of probability theory. In other words, mathematical statistics prepares itself to clarify

the structure of probabilistic-statistical models based on actual observations of various

events. While it is difficult to forecast the general principles governing the behavior

of a small set of random variables, the laws of large numbers encapsulate the notion

that large sets of random variables tend to lose various aspects of their randomness.

8

They stabilize, patterns emerge, and their general behavior becomes fairly predictable.

Indeed, the law of large numbers provides a rigorous mathematical description for

the statistical laws abstracted from the empirical observation that the average of the

results obtained from a large number of trials should be close to a fixed value (called the

expected value or mean in probability theory and mathematical statistics), and will tend

to become closer as more trials are performed.

The first special form of the law of large numbers with rigorous mathematical proof

was given by the prominent Swiss mathematician Jacob Bernoulli in his renowned

book Ars Conjectandi (The Art of Conjecturing) published posthumously in 1713. His

theorem is presently called the weak law of large numbers for Bernoulli trials. According

to Bernoulli’s theorem, if Sn is the number of occurrences of an event A in n independent

trials and p the constant probability of occurrence of event A in each of the independent

trials, then for all positive real numbers ε,

limn→∞P

∣∣∣∣Snn − p∣∣∣∣ < ε

= 1;

that is, in probability terminology, Sn/n converges to p in probability. This theorem was

extended in the next 100 years or so by the great French mathematician and physicist

Simeon D. Poisson and the eminent Russian mathematician Pafnuty L. Chebychev. It

was Poisson who coined the phrase “the law of large numbers” (in French, “la loi des

grands nombres”). In 200 years or so after Bernoulli’s weak law of large numbers, the

French mathematician Emile Borel obtained the strong law of large numbers (SLLN)

for Bernoulli trials, which concludes in probability terminology that Sn/n converges to p

almost certainly (a. c.); that is,

P

limn→∞

Snn= p

= 1.

In 1933, the preeminent Soviet Russian mathematician Andrey N. Kolmogorov

inaugurated the modern era in probability theory in his classic monograph Foundations

9

of the Theory of Probability. Kolmogorov there successfully gives probability theory a

rigorous axiomatic basis, harnessing the full power of measure theory by regarding a

probability event function as a measure of mass one defined on the σ-algebra of events.

His SLLN declares that, for a sequence of i.i.d. random variables X1,X2, ... and a real

number µ, the following are equivalent:

(i) The expected value of X1 exists and is µ; that is, E |X1| < ∞ and EX1 = µ.

(ii) The sample mean converges to µ with probability one; that is,

1

n

n∑i=1

Xi → µ a. c.

This completes the line of work started by Jacob Bernoulli; i.e., the preceding Bernoulli’s

weak law of large numbers. Kolmogorov’s SLLN is the precise form of the folklore idea

of the law of averages, and shows convincingly that the Kolmogorov’s axiomatic system

has successfully captured the true essence of probability theory. Kolmogorov’s SLLN

was extended by Marcinkiewicz-Zygmund (1937) and Feller (1946) who proved SLLNs

for i.i.d. random variables using more general norming sequences.

The classic SLLNs can be extended in various directions and provides intuition

for many other theories. Some SLLNs can be obtained under weakened assumptions,

such as for random variables which are independent but not necessarily identically

distributed, or for random variables which are pairwise independent (Chow and Teicher

(1997, Section 5.2)). Some can hold in more general forms, for example, weighted sums

of random variables. Stout (1974, Chapter 4) gives an excellent survey of known results

up to 1974 on the SLLN problem for weighted sums of independent random variables.

Martingale theory has SLLN type theorems derived via Kolmogorov’s inequality (Feller

(1971, Sections VII.8 and VII.9)). Ergodic theory, motivated by problems of statistical

physics, has its foundations with the SLLN type theorems. The underlying idea is that

for certain systems the time average of their properties converges to the average over

the entire space (the so-called ensemble average). Two of the most important examples

10

are pointwise ergodic theorems of Birkhoff and von Neumann (Shiryaev (1996, Chapter

V)). In mathematical statistics, the preceding SLLN type theorems provide numerous

consistent estimators and statistics.

1.2 Strong Law of Large Numbers for Banach Space Valued Random Elements

In the early 1950s, “Probability in Banach Spaces”, as a branch of modern

mathematics, was initiated by the consideration of a stochastic process as a random

element in a function space (a measurable function from a probability space to a

function space) and, in particular, with the pioneering work by Fortet and Mourier (1953)

on the law of large numbers and the central limit theorem for sums of independent

identically distributed Banach space valued random variables (henceforth to be referred

to as random elements). All technical definitions mentioned in Sections 1.2 and 1.3 will

be reviewed in Chapter 2.

The laws of large numbers for identically distributed (real-valued) random variables

were extended to normed linear spaces by Mourier (1953) and Taylor (1972). Mourier

(1953) established an analogue of the classical Kolmogorov’s SLLN. Specifically,

Mourier showed that, for a sequence of i.i.d. random elements Vn, n ≥ 1 in a real

separable Banach space, if the expected value of V1, denoted by EV1, exists (the

expected value of a random element is defined to be its Pettis integral), then

1

n

n∑i=1

(Vi − EV1)→ 0 a. c.

Taylor (1972) provided conditions for identically distributed random elements in normed

linear spaces to obey the weak law of large numbers.

To obtain the corresponding results for the non-identically distributed random

elements, additional conditions on the distributions of the random elements and/or

on the Banach space itself are needed. A decisive step to the modern development

of probability in Banach spaces was the introduction by Beck (1962) of a convexity

condition on normed linear spaces equivalent to the validity of the extension of a

11

classical SLLN of Kolmogorov. Hoffmann-Jørgensen and Pisier (1976) established

a SLLN by assuming the underlying real separable Banach space is of Rademacher

type p (1 ≤ p ≤ 2). Actually, they showed that for a sequence of independent random

elements Vn, n ≥ 1 with zero expected values in a real separable Banach space, the

Banach space is of Rademacher type p (1 ≤ p ≤ 2) if and only if the following holds:

∞∑n=1

E∥Vn∥p

np< ∞ implies

1

n

n∑i=1

Vi → 0 a. c.

Thus Hoffmann-Jørgensen and Pisier (1976) provided an actual characterization of

Rademacher type p (1 ≤ p ≤ 2) Banach spaces in terms of SLLN. A detailed discussion

may be found in Taylor (1978, Chapter IV).

The study of the SLLN for weighted sums of independent random variables

contributes much to its extension to the SLLN for weighted sums of independent

random elements. Adler and Rosalsky (1987a) and (1987b) presented a general SLLN

for weighted sums of stochastically dominated random variables, which is general

enough to include, as a special case, Feller’s (1946) celebrated extension of the

Marcinkiewicz-Zygmund SLLN (e.g., Chow and Teicher (1997, p. 125)). Adler and

Rosalsky (1987a) did not require the summands to be independent. The hypotheses

involve both the behavior of the tail of the distribution of the dominating random variable

and the growth behavior of the weights and norming constants. Furthermore, the

centering sequence is random. A result of Adler and Rosalsky (1987b) for weighted

sums of i.i.d. random variables is substantially improved by Sung (1997) who obtained

the same theorem but under less stringent conditions.

Based on the work of Adler and Rosalsky (1987a) and (1987b), Adler, Rosalsky,

and Taylor (1989) established a SLLN for weighted sums of independent random

elements in normed linear spaces. The hypotheses involve the distributions of the

independent random elements, the growth behaviors of the weights and norming

constants, and for some of the results a geometric condition is imposed on the normed

12

linear space. Moreover, Adler, Rosalsky, and Taylor (1989) showed that Feller’s (1946)

famous result generalizing the Marcinkiewicz-Zygmund SLLN holds for random elements

in a real separable Rademacher type p (1 < p ≤ 2) Banach space.

Adler, Rosalsky, and Taylor (1992) extended the work of Adler, Rosalsky, and Taylor

(1989) to the case of random weights. They also obtained a SLLN under a uniform

integrability type condition instead of under the geometric condition that the space is of

Rademacher type p (1 < p ≤ 2). Moreover, they established a SLLN for weighted sums

of random elements in real separable semi-normed linear spaces which improves one of

the earlier results of Adler, Rosalsky, and Taylor (1989).

Furthermore, Adler, Rosalsky, and Taylor’s (1989) extension to a Banach space

setting of Feller’s (1946) famous generalization of the Marcinkiewicz-Zygmund SLLN

is obtained as a special case of a very general result of Cantrell and Rosalsky (2002).

Therein, necessary and, separately, sufficient conditions are provided for a sequence

of independent random elements to obey a SLLN. No conditions are imposed on the

underlying Banach space for the necessity result, but for the sufficiency result, it is

assumed that the Banach space is of Rademacher type p (1 ≤ p ≤ 2). Moreover, their

necessity result extends to Banach space setting a result of Martikainen (1979) obtained

for the random variable case and the sufficiency result also includes a well-known

SLLN due to Heyde (1968) for the random variable case. Cantrell and Rosalsky (2004)

established a SLLN for a sequence of independent random elements satisfying a

uniform integrability type condition where no additional conditions are imposed on

the underlying Banach space. Their main result includes as corollaries the SLLN of

Adler, Rosalsky, and Taylor (1992) for a sequence of independent random elements

satisfying a uniform integrability type condition and the SLLN of Taylor and Wei (1979)

for a uniformly tight sequence of independent random elements.

The laws of large numbers in Banach spaces provide powerful tools for many

problems in stochastic process, decision theory, quality control, and statistical estimation

13

theory. Since some stochastic processes can be regarded as being a random element

in particular function spaces, the laws of large numbers for random elements may

be applied. In decision theory, the laws of large numbers can be applied to develop

consistent statistical decision procedures. Quality control is an important industrial

application of statistics. Consistent estimators of the parameters in a continuous

production process can be constructed by using a law of large numbers for weighted

sums of random elements. In the density estimation problem, the intuitive frequency

histogram idea can be extended to a function space approach, and the laws of large

numbers in Banach space can be applied under suitable conditions. A detailed

discussion may be found in Taylor (1978, Chapter VIII).

1.3 Motivation and Organization of Dissertation

Let Vn, n ≥ 1 be a sequence of random elements defined on a probability space

(Ω,F ,P) taking values in a real separable Banach space with norm ∥ · ∥. Suppose that

EVn exists for all n ≥ 1. Let an, n ≥ 1 and bn, n ≥ 1 be sequences of constants

with 0 < bn ↑ ∞. Then an(Vn − EVn) is said to obey the general SLLN with norming

constants bn, n ≥ 1 if the normed weighted sum∑ni=1 ai(Vi − EVi)/bn converges

almost certainly to 0 (the identity of the Banach space as an abelian group under

addition), and this will be written as∑ni=1 ai(Vi − EVi)

bn→ 0 a.c. (1.1)

This dissertation deals with two broad cases:

(i) obtaining SLLNs assuming the Vn, n ≥ 1 are independent and where we imposea condition on the underlying Banach space,

(ii) obtaining SLLNs irrespective of the joint distributions of the Vn, n ≥ 1 and whereno condition is imposed on the underlying Banach space.

The work of part (i), which is established in Chapter 3, is inspired by Sung’s (1997)

extension for Theorem 2 of Adler and Rosalsky (1987b) (Proposition 1.3.1 below).

14

Adler and Rosalsky (1987a) establish some SLLNs for weighted sums of random

variables under rather general conditions. Therein, it is not assumed that the underlying

random variables are independent or identically distributed or even integrable. Adler

and Rosalsky (1987b) in their follow-up article provide sets of necessary and/or

sufficient conditions for the SLLN to hold for weighted sums formed from sequences

of independent and identically distributed (i.i.d.) random variables. In particular, Fernholz

and Teicher’s (1980) main theorem is a special case of Adler and Rosalsky’s (1987b)

Theorem 2 (Proposition 1.3.1 below) taking an = 1, bn = ϕ(dn) and cn = EXn for n ≥ 1

where ϕ is a function defined for positive x such that ϕ(x)/xβ is decreasing for some

β > 1 and 0 < dn ↑ ∞ is a sequence of real numbers satisfying dn+1/dn → 1.

Proposition 1.3.1 (Theorem 2 of Adler and Rosalsky (1987b)). Let Xn, n ≥ 1

be a sequence of i.i.d. Lq random variables for some 1 ≤ q < 2. Let an, n ≥ 1 and

bn, n ≥ 1 be sequences of constants with 0 < bn ↑ ∞,

anbn= O

(1

n1/q

), (1.2)

andn∑i=1

|ai | = O(bn). (1.3)

Then the SLLN ∑ni=1 ai(Xi − EXi)

bn→ 0 a.c. (1.4)

holds.

Sung’s (1997) second result, which is stated in Proposition 1.3.2 below, improves

Theorem 2 of Adler and Rosalsky (1987b) (Proposition 1.3.1).

Proposition 1.3.2 (Theorem 2 of Sung (1997)). Let Xn, n ≥ 1 be a sequence

of i.i.d. Lq random variables for some 1 ≤ q < 2. Let an, n ≥ 1 and bn, n ≥ 1 be

sequences of constants with 0 < bn ↑ ∞. Assume that condition (1.2) holds. Then

(i) (1.4) holds if 1 < q < 2,(ii) (1.4) can fail if q = 1 and condition (1.3) fails.

15

Sung (1997) thus improved Theorem 2 of Adler and Rosalsky (1987b) (Proposition

1.3.1) by showing that condition (1.3) is not needed when 1 < q < 2, and that condition

(1.3) is essential when q = 1. Sung (1997) established part (ii) of Theorem 2 of

Sung (1997) (Proposition 1.3.2) via an example; that is, Sung (1997) gave an example

showing that condition (1.3) cannot be dispensed with in Theorem 2 of Adler and

Rosalsky (1987b) (Proposition 1.3.1) when q = 1.

On the other hand, Theorem 6 of Adler, Rosalsky, and Taylor (1989), which is

stated in Proposition 1.3.3 below, extends Theorem 2 of Adler and Rosalsky (1987b)

(Proposition 1.3.1) from the real line, which is of Rademacher type 2, to a Rademacher

type p (1 < p ≤ 2) Banach space.

Proposition 1.3.3 (Theorem 6 of Adler, Rosalsky, and Taylor (1989)). Let 1 < p ≤ 2

and let Vn, n ≥ 1 a sequence of independent random elements in a real separable

Rademacher type p Banach space X . Suppose that Vn, n ≥ 1 is stochastically

dominated by a random element V in the sense that for some constant D < ∞,

P∥Vn∥ > t ≤ DP∥DV ∥ > t, t ≥ 0, n ≥ 1.

Moreover, suppose that E∥V ∥q < ∞ for some 1 < q < p. Let an, n ≥ 1 and bn, n ≥ 1

be sequences of constants satisfying 0 < bn ↑ ∞ and conditions (1.2) and (1.3). Then

the SLLN (1.1) holds.

Therefore, we were motivated to extend part (i) of Sung’s (1997) second theorem

by obtaining an improved version of Theorem 6 of Adler, Rosalsky, and Taylor (1989).

Indeed we obtained in Theorem 3.2.1, the main result in Chapter 3, an improved version

of Theorem 1 of Adler, Rosalsky, and Taylor (1989) (Proposition 3.2.1 in Chapter 3).

Theorem 3.2.1 readily provides in Theorem 3.2.2 the desired improved version of

Theorem 6 of Adler, Rosalsky, and Taylor (1989).

In our main result, Theorem 3.2.1, we impose the crucial geometric condition on

the real separable Banach space that it is of Rademacher type p (1 ≤ p ≤ 2). De

16

Acosta (1981) established in his Theorem 4.1 the following Marcinkiewicz-Zygmund type

SLLN characterization for a real separable Banach space being of Rademacher type p

(1 ≤ p < 2). The implication ((i)⇒(ii)) was also obtained by Azlarov and Volodin (1981).

Proposition 1.3.4 (Theorem 4.1 of de Acosta (1981)). Let 1 ≤ p < 2. Let X be a

real separable Banach space. Then the following are equivalent:

(i) The Banach space X is of Rademacher type p.

(ii) For every sequence of i.i.d. random elements Vn, n ≥ 1 in X with E∥V1∥p < ∞,∑ni=1(Vi − EVi)n1/p

→ 0 a. c. (1.5)

However, de Acosta (1981) did not provide an explicit example wherein the SLLN

fails for a Banach space which is not of Rademacher type p. This motivated us to

take advantage of Example 7.11 of Ledoux and Talagrand (1991, p. 190) (Example

3.2.2 in Chapter 3). We also noted in Remark 3.2.6 (iv) that in the special case where

Vn, n ≥ 1 is a sequence of i.i.d. random elements with E∥V1∥q < ∞, an = 1, bn = n1/q,

n ≥ 1 where 1 < q < p ≤ 2 and the underlying Banach space is of Rademacher type p,

Theorem 3.2.2 reduces to the Marcinkiewicz-Zygmund type SLLN∑ni=1(Vi − EVi)n1/q

→ 0 a. c. (1.6)

of Woyczynski (1980, Theorem 4.1). This result of Woyczynski (1980) of course does

not follow from Theorem 6 of Adler, Rosalsky, and Taylor (1989) because condition

(1.3) does not hold. By Theorem 4.1 of de Acosta (1981) (Proposition 1.3.4) or by the

cited result of Azlarov and Volodin (1981), the Marcinkiewicz-Zygmund type SLLN (1.6)

holds under the assumption that the Banach space X is of Rademacher type q, which is

weaker than X being Rademacher type p. Indeed, Example 3.2.2 shows explicitly that

the Marcinkiewicz-Zygmund SLLN (1.6) can fail in a Banach space setting without the

Rademacher type p hypothesis. Inspired by Beck (1963), we also construct in Example

17

3.2.3 a sequence of independent but not identically distributed random elements to

serve the same purpose apropos of Theorem 3.2.1.

As was mentioned above, de Acosta (1981) provided a characterization in his

Theorem 4.1 of Rademacher type p (1 ≤ p < 2) Banach spaces via a Marcinkiewicz-

Zygmund type SLLN. The key result used by de Acosta (1981) to prove the SLLN (1.5)

is the following result of de Acosta (1981, Theorem 3.1).

Proposition 1.3.5 (Theorem 3.1 of de Acosta (1981)). Let 1 ≤ p < 2 and let X

be a real separable Banach space. Then for every sequence of i.i.d. random elements

Vn, n ≥ 1 in X with E∥V1∥p < ∞, the SLLN (1.5) holds if and only if∑ni=1(Vi − EVi)n1/p

P→ 0. (1.7)

De Acosta (1981, Theorem 3.1) and de Acosta (1981, Theorem 4.1) together assert

that Rademacher type p (1 ≤ p < 2) Banach spaces can be characterized by the

Marcinkiewicz-Zygmund type weak law of large numbers (1.7).

In summary, we establish in Chapter 3 the work of part (i) obtaining SLLNs

assuming the Vn, n ≥ 1 are independent and the underlying Banach space is of

Rademacher type p (1 ≤ p ≤ 2). Moreover, Theorem 3.2.1 and Theorem 3.2.2 are new

results even when the underlying Banach space is the real line R.

The work of part (ii), which is presented in Chapter 4, is a parallel development of

the work of part (i) presented in Chapter 3 but the arguments are distinctly different.

With the work of part (i) in hand, it seemed natural to develop a “0 < q < 1”

version of Theorem 3.2.2. Note that there are two cases for the real line version of the

Marcinkiewicz-Zygmund SLLN: the condition for random variables being Lq integrable

for 1 ≤ q < 2 and for 0 < q < 1. Since Sawyer (1966), Chatterji (1970), and Martikainen

and Petrov (1980) demonstrated that the real line version of the Marcinkiewicz-Zygmund

SLLN holds without the independence hypothesis when 0 < q < 1, we were inspired to

dispense with the independence assumption. We do not even impose further conditions,

18

such as a Rademacher type p (1 < p ≤ 2) condition, on the underlying Banach space.

We obtain in Theorem 4.2.1, the first main result of Chapter 4, a SLLN of the form∑ni=1 aiVi

bn→ 0 a.c. (1.8)

which is parallel to the 0 < q < 1 part of the real line version of the Marcinkiewicz-Zygmund

SLLN.

We now recall the real line version Marcinkiewicz-Zygmund SLLN, which generalizes

the classical Kolmogorov’s SLLN as was mentioned in Section 1.1. Its proof may be

found in Chow and Teicher (1997, p. 125).

Proposition 1.3.6 (Real line version Marcinkiewicz-Zygmund SLLN). Let Xn, n ≥

1 be a sequence of i.i.d. random variables and let 0 < q < 2. Then∑ni=1 Xi − ncn1/q

→ 0 a. c.

for some finite constant c if and only if E |X1|q < ∞. In such a case, c = EX1 if 1 ≤ q < 2

while c is arbitrary (and hence may be taken as zero) if 0 < q < 1.

Theorem 4.2.1 establishes a SLLN of the form (1.8) for a sequence of Banach

space valued random elements Vn, n ≥ 1 which is stochastically dominated by a

random element V with E∥V ∥q < ∞ for some 0 < q < 1. The conclusion (1.8) holds

irrespective of the joint distributions of the Vn, n ≥ 1. The real line version of the

Marcinkiewicz-Zygmund SLLN has an ≡ 1 and bn = n1/q, n ≥ 1. In Theorem 4.2.1 we

impose the condition

anbn= O

(1

n1/q

). (1.9)

which is of course automatic if an ≡ 1 and bn = n1/q, n ≥ 1.

19

Petrov (1973, Theorem 1), Rosalsky and Stoica (2010, Theorem 2.1), and Rosalsky

and Stoica (2010, Theorem 2.2) obtained real line SLLNs of the form∑ni=1 Xi

bn→ 0 a. c.

irrespective of the joint distributions of the random variables Xn, n ≥ 1. We were thus

enlightened to extend their results to the general form (1.8) for Banach space valued

random elements.

Petrov (1973, Theorem 1) motivated us to replace condition (1.9) by the condition

∞∑n=1

(|an|bn

)q< ∞ (1.10)

under which we obtain a SLLN in Theorem 4.2.2, our second main result in Chapter 4.

When |an|/bn ↓, (1.10) is stronger than (1.9). Though the 0 < q < 1 part of Theorem

4.2.2 is thus a direct corollary of Theorem 4.2.1, Theorem 4.2.2 extends Theorem 4.2.1

in the sense that it can produce the SLLN (1.8) even when q = 1. Theorem 4.2.2 is new

even when the Banach space is the real line and as a corollary it yields Theorem 1 of

Petrov(1973).

Inspired by Rosalsky and Stoica (2010, Theorem 2.1), and Rosalsky and Stoica

(2010, Theorem 2.2), we obtain the SLLN (1.8) under different conditions in Theorem

4.2.3 and Theorem 4.2.4, respectively. Theorems 4.2.3 and 4.2.4 are also new even

when the underlying Banach space is the real line, and they can incorporate as

corollaries the results of Rosalsky and Stoica (2010, Theorem 2.1) and Rosalsky

and Stoica (2010, Theorem 2.2), respectively.

While the Banach space SLLN results obtained in the current work are new even

in the random variable case, some corollaries or special cases of some of the random

element (and random variable) results are well known as we have discussed above.

Consequently, the current work is a bona fide extension of previously established

results. Illustrative examples are provided throughout to compare the results or

20

show how the results improve upon or are different from other results in the literature.

Examples are also provided to show that the results are sharp. Prior to the presentation

of the main results in Chapters 3 and 4, notation, definitions, and some relevant results

about Banach spaces are presented in Chapter 2, Section 1. Probabilistic concepts in

Banach spaces are presented in Chapter 2, Section 2. Chapter 2, Section 3 contains the

lemmas needed in Chapters 3 and 4.

We end this chapter by mentioning that mean convergence versions of the

Marcinkievicz-Zygmund SLLN have been investigated for both the cases of a sequence

of random variables and a sequence of Banach space valued random elements. Klass

(1973, Corollary 12) proved for a sequence of i.i.d. random variables Xn, n ≥ 1 with

E |X1|p < ∞ for some p in [1, 2) that

limn→∞

E |∑ni=1(Xi − EXi)|n1/p

= 0.

Korzeniowski (1984) extended this result of Klass (1973) to the case of a sequence of

random elements taking values in a real separable Rademacher type q (1 < q ≤ 2)

Banach space X (which is automatic if X = R). Specifically, it follows from Theorem

2 of Korzeniowski (1984) that for a sequence of i.i.d. random elements Vn, n ≥ 1

taking values in a real separable Banach space which is of Rademacher type q for some

1 < q ≤ 2, if E∥V1∥p < ∞ for some 1 ≤ p < q, then

limn→∞

E ∥∑ni=1(Vi − EVi)∥n1/p

= 0.

However, in this dissertation, we will concentrate on obtaining results for the a.c.

convergence to 0 of normed partial sums and we will not obtain mean convergence

results.

21

CHAPTER 2PRELIMINARIES: DEFINITIONS, LEMMAS, AND NOTATION

2.1 Basic Concepts of Banach Spaces

Some definitions, lemmas, and notation need to be presented prior to stating and

proving the main results.

A nonempty set X is said to be a (real) linear space if there is defined a binary

operation of addition which makes X an abelian group and an operation of multiplication

by (real) scalars which satisfy the distributive and identity laws; this is stated more

precisely as follows.

(a) To every pair of element (u, v) ∈ X × X , there corresponds an element w ∈ Xsuch that w = u + v .

(b) To every u ∈ X and t ∈ R, there corresponds an element tu ∈ X .

(c) The operations defined in (a) and (b) satisfy, for all u, v ,w ∈ X and all s, t ∈ R, thefollowing seven properties:

(i) u + v = v + u,(ii) (u + v) + w = u + (v + w),(iii) u + v = u + w implies v = w ,(iv) 1u = u,(v) (st)u = s(tu),(vi) (s + t)u = su + tu,(vii) s(u + v) = su + sv .

The zero element of X is denoted by 0. While this is the same symbol as the real

number 0, it should be clear from the context as to whether 0 refers to 0 ∈ X or 0 ∈ R.

A real linear space X is said to be normed if there is a real-valued function defined

on X and denoted by ∥ · ∥ such that ∥ · ∥ satisfies, for all u, v ∈ X and all t ∈ R, the

following three properties:

(i) ∥u∥ ≥ 0 and ∥u∥ = 0 if and only if u = 0,

(ii) ∥u + v∥ ≤ ∥u∥+ ∥v∥,

(iii) ∥tu∥ = |t| · ∥u∥.

22

The function ∥ · ∥ is then called a norm on X . Property (ii) above is called the triangle

inequality.

A sequence vn, n ≥ 1 in a normed linear space X is said to converge to an

element v of X if limn→∞

∥vn − v∥ = 0. This will be denoted by limn→∞vn = v or by vn → v

as n → ∞. A sequence vn, n ≥ 1 in a normed linear space X is said to be a Cauchy

sequence if for every ε > 0, there exists an integer N such that ∥vn − vm∥ < ε whenever

n ≥ N and m ≥ N; i.e.,

limn→∞

supm>n

∥vm − vn∥ = 0.

A normed linear space X is said to be complete if every Cauchy sequence of X

converges to an element of X . A complete normed linear space is called a Banach

space.

A subset S of a normed linear space X is said to be dense in X if its closure (that

is, the smallest closed subset of X containing S) equals X . If X has a countable dense

subset, then X is said to be separable.

In the following examples we list several particular real Banach spaces.

Example 2.1.1. The space ℓp, 1 ≤ p < ∞, is the class of all real sequences

v = (v1, v2, ...) such that∑∞k=1 |vk |p < ∞. With the norm defined by

∥v∥p =

(∞∑k=1

|vk |p)1/p

,

each of the spaces ℓp, 1 ≤ p < ∞, is a real separable Banach space.

Example 2.1.2. The space ℓ∞ is the collection of all bounded real sequences

v = (v1, v2, ...). With the norm defined by

∥v∥∞ = sup|vk |, k ≥ 1,

ℓ∞ is a real Banach space which is not separable (e.g., Taylor (1978, p. 10)). Let c0

denote the subspace of ℓ∞ which consists of the real sequences that converge to zero.

With the same norm as ℓ∞, c0 is a real separable Banach space.

23

Example 2.1.3. The space Lp(R), 1 ≤ p < ∞, is the class of all real Lebesgue

measurable functions v(·) on R such that∫R |v(t)|

p dt < ∞. With the norm defined by

∥v∥p =(∫

R|v(t)|p dt

)1/p,

each of the spaces Lp(R), 1 ≤ p < ∞, is a real separable Banach space.

Example 2.1.4. The space L∞(R) is the class of all real Lebesgue measurable

functions v(·) that are bounded almost everywhere (a.e.) on R with respect to Lebesgue

measure. With the norm defined by

∥v∥∞ = infδ : |v(t)| ≤ δ a.e.,

the space L∞(R) is a Banach space which is not separable (e.g., Taylor (1978, p. 11)).

The norm ∥v∥∞ is called the essential supremum of |v(·)| and is also denoted by β(|v |).

The collection of all continuous linear functionals (that is, continuous real-valued

linear functions) defined on a normed linear space X is called the dual space of X and

is denoted by X ∗. We recall that a linear functional is a function f : X → R satisfying

f (au + bv) = af (u) + bf (v) for all u, v ∈ X and all a, b ∈ R.

A sequence bn, n ≥ 1 in a Banach space X is said to be a Schauder basis for X if

for each v ∈ X there exists a unique sequence of scalars tn, n ≥ 1 such that

v = limn→∞

n∑k=1

tkbk .

When X has a Schauder basis bn, n ≥ 1, a sequence of linear functionals fk , k ≥ 1

can be defined by

fk(v) = tk , k = 1, 2, ...

where v ∈ X and v = limn→∞

∑nk=1 tkbk . The linear functionals fk , k ≥ 1 ⊆ X ∗ are called

the coordinate functionals.

24

The following Theorem 2.1.1 is the Riesz Representation Theorem (e.g., Royden

(1988, p. 132)) and it will be used in Example 2.2.3 below. The Riesz Representation

Theorem is a crowning achievement in twentieth century mathematics.

Theorem 2.1.1 (Riesz Representation Theorem). For each f in the dual space of

Lp(R), 1 ≤ p < ∞, there exists gf ∈ Lq(R) where 1/p + 1/q = 1 (q = ∞ if p = 1) such

that

f (h) =

∫Rh(x)gf (x)dx for all h ∈ Lp(R).

Remark 2.1.1. The Riesz Representation Theorem is termed a “representation”

theorem because it provides a concrete “representation” for the members of the dual

space of Lp(R), 1 ≤ p < ∞. Informally, Theorem 2.1.1 asserts that the dual space of

Lp(R), 1 ≤ p < ∞ is Lq(R) where 1/p + 1/q = 1 (q =∞ if p = 1).

2.2 Probability in Banach Spaces

Let (Ω,F ,P) be a probability space. Let X denote a real separable Banach space

with a norm ∥ · ∥. Let X be equipped with its Borel σ-algebra B(X ); i.e., B(X ) is the

σ-algebra generated by the class of open subsets of X determined by the metric

d(u, v) = ∥u− v∥, u, v ∈ X . A random element V in X is a F-measurable transformation

from Ω to the measurable space (X ,B(X )); i.e., V−1(A) ∈ F for all A ∈ B(X ).

Remark 2.2.1. A random element is a generalization of a random variable since

the Borel σ-algebra generated by all intervals of real numbers of the form (−∞, b) is the

class of Borel subsets of R. Therefore, V is a random element in R if and only if V is a

random variable. Furthermore, random elements in an n-dimensional Euclidean space

Rn are n-dimensional random vectors.

The following Proposition 2.2.1 shows that some properties of random variables can

be extended to the setting of random elements. A further discussion may be found in

Taylor (1978, Chapter II).

25

Proposition 2.2.1 (Taylor (1978)). (i) Let Vn, n ≥ 1 be a sequence of randomelements in a Banach space X such that Vn(ω) converges to V (ω) for each ω ∈ Ω.Then V is a random element in X .

(ii) Let V be a random element in a Banach space X and let Y be a randomvariable. Then YV is a random element in X .

(iii) If the real Banach space X is separable, then ∥V −W ∥ is a random variablewhenever V andW are random elements in X . In particular, takingW = 0, ∥V ∥ isa random variable if V is a random element.

(iv) If the Banach space X is separable, then a function V : Ω → X is a randomelement in X if and only if f (V ) is a random variable for each f ∈ X ∗.

Remark 2.2.2. (i) The necessity half in Proposition 2.2.1 (iv) is true withoutthe assumption that X is separable.

(ii) Not all of the properties of random variables can be extended to the settingof random elements. For example, the sum of two random variables are randomvariables, but the sum of two random elements in a Banach space X may not bemeasurable. However, if X is separable, then we see from Proposition 2.2.1 (iv)that the sum of two random elements in X is a random element in X .

(iii) Taylor (1978, p. 26) presented an example showing that if X is not separable,then ∥V −W ∥ is not necessarily a random variable where V ,W , and V −W arerandom elements in X . Consequently, Proposition 2.2.1 (iii) can fail without theassumption that X is separable.

We now define modes of convergence of a sequence of random elements in a real

separable Banach space. Let Vn, n ≥ 1 be a sequence of random elements in a real

separable Banach space X . Then Vn, n ≥ 1 converges to a random element V in X

(i) with probability one or almost certainly (a. c.) if Plimn→∞

∥Vn − V ∥ = 0= 1, and

this is denoted Vn → V a. c. (or limn→∞Vn = V a. c.).

(ii) in probability if limn→∞P∥Vn − V ∥ ≥ ε = 0 for all ε > 0, and this is denoted Vn

P→ V .

(iii) in the r th mean for r > 0 if E∥Vn∥r < ∞ for all n ≥ 1 and limn→∞E∥Vn − V ∥r = 0, and

this denoted VnLr→ V . Necessarily, we have E∥V ∥r < ∞.

26

A random element in a Banach space and the underlying probability measure

induce a probability measure on the Banach space and its Borel subsets. The proba-

bility distribution of a random element V in a Banach space X is the induced measure,

denoted by PV , on (X ,B(X )); i.e.,

PV B = PV ∈ B, B ∈ B(X ).

The random elements V andW in X are said to be identically distributed if

PV ∈ B = PW ∈ B for all B ∈ B(X ).

A family of random elements in X is said to be identically distributed if its every pair is

identically distributed. A finite set of random elements V1, ... ,Vn in X is said to be

independent if for every choice of B1, ... ,Bn ∈ B(X ),

PV1 ∈ B1, ... ,Vn ∈ Bn = PV1 ∈ B1 · · ·PVn ∈ Bn.

A family of random elements in X is said to be independent if its every finite subset is

independent.

The expected value or mean of a random element V in a real separable Banach

space X , denoted EV , is defined to be the Pettis integral provided it exists; i.e., V has

the expected EV in X if for each f ∈ X ∗, we have

E [f (V )] = f (EV ) (2.1)

where X ∗ is the dual space of X . Note that the left-hand side of (2.1) makes sense

because of Proposition 2.2.1 (iv) and also note that necessarily f (V ) is integrable for

each f ∈ X ∗. The Pettis integral was introduced by Pettis in 1938 (Pettis (1938)). A

complete characterization of when the Pettis integral exists was provided by Brooks

(1969). A further discussion and details regarding the properties of the Pettis integral

may be found in Hille and Phillips (1985, pp. 76–85).

27

The expected value of random elements enjoys similar properties as does the

expected value of random variables (Proposition 2.2.2 below) and sometimes can be

obtained as in the random variable case (Proposition 2.2.3 and Example 2.2.3 below).

We illustrate the definition of the expected value of a random element with the

following very simple example (Example 2.2.1). More involved examples (Examples

2.2.2 and 2.2.3) are presented below.

Example 2.2.1. Let X be an L1 random variable and let v ∈ X where X is an

arbitrary real separable Banach space. Let V = Xv . Then the expected value EV of V

exists and is given by EV = (EX )v . (This is of course precisely what one would expect

to be the expected value of V .)

Proof : By Proposition 2.2.1 (ii), V = Xv is a random element in X since v can be

regarded as a degenerate random element in X . Then f (V ) = f (Xv) = Xf (v) for all

f ∈ X ∗ since X (ω) can be regarded as a real scalar for each ω ∈ Ω. Thus, (2.1) holds

since E [f (V )] = E [Xf (v)] = f (v)EX = f ((EX )v) for all f ∈ X ∗. Therefore, the expected

value EV of V exists and is given by EV = (EX )v . 2

Proposition 2.2.2 (Taylor (1978)). Let V , V1 and V2 be random elements in a real

separable Banach space X , then

(i) If EV1 and EV2 exist, then E(V1 + V2) exists and E(V1 + V2) = EV1 + EV2.

(ii) If EV exists and t ∈ R, then E(tV ) exists and E(tV ) = tEV .

(iii) If E∥V ∥ < ∞, then the expected value EV of V exists and ∥EV ∥ ≤ E∥V ∥.

Proposition 2.2.3. If V is a countably-valued random element in X taking values

vi , i ≥ 1, then the expected value EV of V exists and is given by

EV =

∞∑i=1

viPV = vi

provided∞∑i=1

∥vi∥PV = vi < ∞.

28

Proof : Let v =∞∑i=1

viPV = vi. Then v ∈ X since X is complete. Moreover, (2.1)

holds since for each f ∈ X ∗,

E [f (V )]

=

∞∑i=1

f (vi)PV = vi

= limn→∞

n∑i=1

f (vi)PV = vi

= limn→∞f

(n∑i=1

viPV = vi

)

=f

(limn→∞

n∑i=1

viPV = vi

)(since f is continuous)

=f

(∞∑i=1

viPV = vi

)

=f (v).

Thus, the expected value of V exists and is given by EV = v =∞∑i=1

viPV = vi. 2

Example 2.2.2. If a Banach space X has a Schauder basis bn, n ≥ 1 with

coordinate functionals fn, n ≥ 1, then each random element V in X can be expressed

as V =∞∑n=1

fn(V )bn pointwise in ω ∈ Ω. If V has expected value EV ∈ X , then

E [fn(V )] = fn(EV ) since each fn is in X ∗. Thus

EV =

∞∑n=1

fn(EV )bn =

∞∑n=1

E [fn(V )]bn. (2.2)

Note that the spaces ℓp, 1 ≤ p < ∞ share the same Schauder basis v (n), n ≥ 1

where v (n) is the element of ℓp having 1 in its nth position and 0 elsewhere. Thus, each

random element V in ℓp, 1 ≤ p < ∞ can be expressed as a sequence of random

variables fn(V ), n ≥ 1; i.e., V = (f1(V ), f2(V ), ...) = (V1,V2, ...) (say). Furthermore, if

29

the expected value EV of V exists, then by (2.2) we get

EV =

∞∑n=1

E [fn(V )]v(n) = (E(f1(V )),E(f2(V )), ...) = (EV1,EV2, ...). (2.3)

(Again, this is precisely what one would expect to be the expected value of V .)

Paralleling the Riesz Representation Theorem which concerns the real separable

Banach space Lp(R), 1 ≤ p < ∞, the following “representation” theorem for ℓp,

1 ≤ p < ∞ (Wilansky (1964, p. 91)) will be used in Remark 2.2.3 below which pertains

to Example 2.2.2.

Theorem 2.2.1. For each f ∈ ℓ∗p, 1 ≤ p < ∞, there exists b(f ) = (b1(f ), b2(f ), ...) ∈

ℓq where 1/p + 1/q = 1 (q =∞ if p = 1) such that

f (a) =

∞∑n=1

anbn(f ) for all a = (a1, a2, ...) ∈ ℓp.

Remark 2.2.3. Let V = (V1,V2, ...) be a random element in ℓp (1 ≤ p < ∞) as in

Example 2.2.2. If we assume

∞∑n=1

E |Vn|p < ∞ (2.4)

(that is, (E |V1|p,E |V2|p, ...) ∈ ℓ1), then we also obtain the expected value of V with the

form (2.3) via Theorem 2.2.1 as follows. Note at the outset that (2.4) implies that Vn is

integrable for each n ≥ 1. Let v = (EV1,EV2, ...). Then v ∈ ℓp since

∥v∥p =

(∞∑n=1

|EVn|p)1/p

≤

(∞∑n=1

E |Vn|p)1/p

(by Jensen’s Inequality)

< ∞ (by (2.4)).

By Theorem 2.2.1, f (V ) =∞∑n=1

Vnbn(f ) for each f ∈ ℓ∗p where

b(f ) = (b1(f ), b2(f )), ...) ∈ ℓq and 1/p + 1/q = 1 (q =∞ if p = 1).

30

Then, for each m ≥ 1,∣∣∣∣∣m∑n=1

Vnbn(f )

∣∣∣∣∣ ≤m∑n=1

|Vn| · |bn(f )| ≤∞∑n=1

|Vn| · |bn(f )|

≤ ∥V ∥p∥b(f )∥q (by Holder’s Inequality).

Moreover, ∥V ∥p∥b(f )∥q is integrable since

E(∥V ∥p∥b(f )∥q)

=∥b(f )∥q E∥V ∥p

=∥b(f )∥q E

( ∞∑n=1

|Vn|p)1/p

≤∥b(f )∥q

[E

(∞∑n=1

|Vn|p)]1/p

(by Jensen’s Inequality)

=∥b(f )∥q

[∞∑n=1

E |Vn|p]1/p

(by Lemma 2.3.6)

<∞ (by (2.4)).

Thus, by the Lebesgue Dominated Convergence Theorem,

E [f (V )] = E

(∞∑n=1

Vnbn(f )

)

= E

(limm→∞

m∑n=1

Vnbn(f )

)

= limm→∞

E

(m∑n=1

Vnbn(f )

)

= limm→∞

(m∑n=1

(EVn)bn(f )

)

=

∞∑n=1

(EVn)bn(f )

= f (v) (by Theorem 2.2.1)

31

recalling that v = (EV1,EV2, ...). Hence, the expected value EV of V exists and is given

by (2.3).

Example 2.2.3. Let V be a random element in X = Lp(R), 1 ≤ p < ∞ (Example

2.1.3) with∫Ω

∫R |V(ω)(x)|

pdxdP(ω) < ∞. Then the expected value of EV of V exists

and is given by EV =∫ΩV dP viewed as a function of x ∈ R; i.e., EV : R → R is given

by x 7→∫ΩV(ω)(x)dP(ω). (Once again, this is precisely what one would expect to be

the expected value of V .) Figure 2-1 below is provided to help clarify the notion of the

expected value of a random element V in Lp(R), 1 ≤ p < ∞.

Figure 2-1. Expected Value of a Random Element in Lp(R), 1 ≤ p < ∞

Proof : For fixed x ∈ R, V(·)(x) is a random variable. Define a function v(·) on R by

v(x) = EV(·)(x) =

∫Ω

V(ω)(x)dP(ω) for all x ∈ R.

Then v(·) is Lebesgue measurable. Since 1 ≤ p < ∞, we have for each x ∈ R that

|v(x)|p = |EV(·)(x)|p ≤ E |V(·)(x)|p by Jensen’s inequality. So v(·) ∈ Lp(R) since∫R|v(x)|pdx ≤

∫RE |V(·)(x)|pdx

32

=

∫R

∫Ω

|V(ω)(x)|pdP(ω)dx

=

∫Ω

∫R|V(ω)(x)|pdxdP(ω)

< ∞.

On the other hand, for fixed ω ∈ Ω, V(ω)(·) ∈ Lp(R). By the Riesz Representation

Theorem (Theorem 2.1.1), for each f in the dual space of Lp(R), there exists gf ∈ Lq(R)

where 1/p + 1/q = 1 (q =∞ if p = 1) such that

f (h) =

∫Rh(x)gf (x)dx for all h ∈ Lp(R).

Therefore, for each f in the dual space of Lp(R), by first taking h(·) = V(ω)(·) and then

taking h(·) = v(·), we get

E [f (V )] =

∫Ω

f (V(ω)(·))dP(ω)

=

∫Ω

(∫RV(ω)(x)gf (x)dx

)dP(ω)

=

∫R

(∫Ω

V(ω)(x)dP(ω)

)gf (x)dx (by Fubini’s Theorem)

=

∫Rv(x)gf (x)dx

= f (v).

Hence, the expected value EV of V exists and is given by EV = v =∫ΩV dP. 2

The following example shows that the expected value EV can exist even if E∥V ∥ =

∞.

Example 2.2.4 (Taylor (1978, p. 41)). For the real separable Banach space ℓ2,

define a random element V such that V = nv (n) with probability c/n2 where v (n) is

the element of ℓ2 having 1 in its nth position and 0 elsewhere and c is an appropriate

constant. Note that

E∥V ∥2 =∞∑n=1

nc

n2= c

∞∑n=1

1

n=∞,

33

However, by Proposition 2.2.3,

EV =

∞∑n=1

nv (n)P(V = nv (n)) =

∞∑n=1

nv (n)c

n2=(c1,c

2, ... ,

c

n, ...)∈ ℓ2.

Let εn, n ≥ 1 be a symmetric Bernoulli sequence; i.e., εn, n ≥ 1 is a sequence

of independent and identically distributed (i.i.d.) random variables with

Pεn = 1 = Pεn = −1 = 1/2, n ≥ 1.

A symmetric Bernoulli sequence is also referred to as a Rademacher sequence. Let

X∞ = X × X ××X × · · · , and define

C(X ) =

(v1, v2, ...) ∈ X∞ :

∞∑n=1

εnvn converges in probability

.

Let 1 ≤ p ≤ 2. Then a real separable Banach space X is said to be of Rademacher type

p if there exists a constant 0 < C < ∞ such that

E

∥∥∥∥∥∞∑n=1

εnvn

∥∥∥∥∥p

≤ C∞∑n=1

∥vn∥p

for all (v1, v2, ...) ∈ C(X ). Hoffmann-Jørgensen and Pisier (1976) proved for 1 ≤ p ≤ 2

that a real separable Banach space is of Rademacher type p if and only if there exists a

constant 0 < C < ∞ such that

E

∥∥∥∥∥n∑i=1

Vi

∥∥∥∥∥p

≤ Cn∑i=1

E∥Vi∥p

for every finite collection V1, ... ,Vn of independent random elements in X with zero

expected values.

If a real separable Banach space is of Rademacher type p for some 1 < p ≤ 2,

then it is of Rademacher type q for all 1 ≤ q < p. Every real separable Banach

space is of Rademacher type (at least) 1 while the Lp-spaces and ℓp-spaces are of

Rademacher type min2, p for p ≥ 1. Every real separable Hilbert space and real

separable finite-dimensional Banach space is of Rademacher type 2; in particular, the

34

real line R is of Rademacher type 2. The real separable Banach space c0 (Example

2.1.2) is not of Rademacher type p for any p ∈ (1, 2] and for q ∈ [1, 2), the real separable

Banach spaces Lq and ℓq are not of Rademacher type p for any p ∈ (q, 2]. A detailed

discussion of the above can be found in Chapter 9 of Ledoux and Talagrand (1991). The

real Banach space ℓ∞ is not even separable as was mentioned in Example 2.1.2.

2.3 Useful Lemmas

The classical and celebrated real line version of Levy’s Theorem (e.g., Chow

and Teicher (1997, p. 72)), which asserts that the partial sums from a sequence of

independent random variables converge almost certainly to a random variable S if and

only if they converge in probability to S , has been extended to a real separable Banach

space setting by Ito and Nisio (1968) and is stated as follows.

Lemma 2.3.1 (Ito and Nisio (1968)). Let Vn, n ≥ 1 be a sequence of independent

random elements in a real separable Banach space X and set

Sn =

n∑i=1

Vi , n ≥ 1.

Then Sn converges a.c. to a random element S in X if and only if SnP→ S .

Remark 2.3.1. It follows from Lemma 2.3.1 that in the definition of C(X ), the

condition∞∑n=1

εnvn converges in probability

is equivalent to the condition

∞∑n=1

εnvn converges a.c.

Now we introduce the notion of regular variation which has been proved fruitful in an

increasing number of applications in probability theory (Feller (1971, VIII.8 and VIII.9) for

a detailed discussion). A positive Borel function L defined on [0,∞) is said to vary slowly

35

(at infinity) (or be slowly varying (at infinity)) if

limx→∞

L(cx)

L(x)= 1 for all c > 0.

A positive Borel function R on [0,∞) is said to vary regularly (or be regularly varying)

with exponent ρ (−∞ < ρ < ∞) if it is of the form R(x) = xρL(x) with L slowly varying;

i.e.,

limx→∞

R(cx)

R(x)= cρ for all c > 0.

Clearly, a function is slowly varying if and only if it is regularly varying with exponent

ρ = 0, and a positive Borel function L is slowly varying if and only if 1/L is slowly varying.

For example, all powers of | log x | are slowly varying. Similarly, a function approaching a

positive finite limit is slowly varying.

Feller (1971) introduced the following two abbreviations:

Zu(x) =

∫ x0

y uZ(y)dy , Z ∗u (x) =

∫ ∞

x

y uZ(y)dy , −∞ < u < ∞, (2.5)

where Z is a regularly varying function, and explored their asymptotic properties as

x → ∞ (Lemmas 2.3.2 and 2.3.3 below). We will apply these properties in Examples

3.2.6, 3.2.8, 3.2.9, and 4.2.4.

Lemma 2.3.2 (Feller (1971, p. 280)). Let Z > 0 vary slowly. Then the integrals

in (2.5) converge at ∞ for u < −1 and diverge for u > −1. If u ≥ −1, then Zu varies

regularly with exponent u + 1. If u < −1, then Z ∗u varies regularly with exponent u + 1,

and this remains true for u = −1 if Z ∗−1 < ∞.

Lemma 2.3.3 (Feller (1971, p. 281)). (i) If Z varies regularly with exponent ρand Z ∗

u < ∞, then u + ρ+ 1 ≤ 0 and

limx→∞

xu+1Z(x)

Z ∗u (x)

= λ

where λ = −(u + ρ+ 1) ≥ 0.

36

(ii) If Z varies regularly with exponent ρ and if u ≥ −(ρ+ 1) then

limx→∞

xu+1Z(x)

Zu(x)= λ

where λ = u + ρ+ 1 ≥ 0.

The Borel-Cantelli lemma (e.g., Chow and Teicher (1997, p. 42)) plays an

indispensable role in probability theory for establishing a.c. convergence results

and is stated as follows. For a sequence of events An, n ≥ 1 we recall that

lim supn→∞

An =∞∩n=1

∞∪k=n

Ak and that lim supn→∞

An is also conveniently denoted by [An i.o. (n)]

where i.o. (n) signifies “infinitely often in n”.

Lemma 2.3.4 (Borel-Cantelli Lemma). If An, n ≥ 1 is a sequence of events for

which∞∑n=1

PAn < ∞, then Plim supn→∞

An

= 0 or, equivalently, P

lim infn→∞

Acn

= 1.

The classical real line version of the Kronecker’s lemma (e.g., Chow and Teicher

(1997, p. 114)) carries over to a Banach space (e.g., Taylor (1978, p. 101)) and this

Banach space version is stated as follows.

Lemma 2.3.5. Let vn, n ≥ 1 be a sequence of elements in a real Banach space

and let bn, n ≥ 1 be a sequence of real positive numbers tending to infinity. If

∞∑n=1

vnbn

converges,

then1

bn

n∑i=1

vi → 0.

The following lemma (Lemma 2.3.6), the Beppo-Levi Theorem (e.g., Chow and

Teicher (1997, p. 90, Corollary 2)), is a direct corollary of the Monotone Convergence

Theorem.

Lemma 2.3.6 (Beppo-Levi). Let Xn, n ≥ 1 be a sequence of nonnegative random

variables on (Ω,F ,P). Then

E

(∞∑n=1

Xn

)=

∞∑n=1

EXn.

37

For a real separable Banach space X and p ∈ [1,∞), it is well known that the

class of random elements V for which E∥V ∥p < ∞ forms a Banach space with norm

(E∥V ∥p)1/p (e.g., Hille and Phillips (1985, p. 89)). We thus have the following result.

Lemma 2.3.7. Let Vn, n ≥ 1 be a sequence of random elements in a real

separable Banach space X and let p ∈ [1,∞). If

E∥Vn∥p < ∞, n ≥ 1

and

limn→∞

supm>nE∥Vm − Vn∥p = 0,

then there exists a random element V in X such that

limn→∞E∥Vn − V ∥p = 0.

Remark 2.3.2. When X is the real line, Lemma 2.3.7 reduces to the well-known

Cauchy convergence criterion for random variables (e.g., Chow and Teicher (1997, p.

99)). The proof of Lemma 2.3.7 given by Hille and Phillips (1985, p. 89) follows along

the lines of the proof of the Cauchy convergence criterion for random variables, mutatis

mutandis.

The following lemma (Lemma 2.3.8), the Levy Central Limit Theorem (e.g., Chow

and Teicher (1997, p. 317)), is a direct corollary of the Lindeberg-Feller Central Limit

Theorem (e.g., Chow and Teicher (1997, p. 314)). Together with the Kolmogorov

Zero-One Law (e.g., Chow and Teicher (1997, p. 64)), we obtain Lemma 2.3.9 and apply

it in Example 3.2.11 below.

Lemma 2.3.8 (Levy Central Limit Theorem). Let Sn =n∑i=1

Xi where Xn, n ≥ 1 is a

sequence of i.i.d. random variables with EX1 = µ, Var(X1) = σ2 ∈ (0,∞). Then

Sn − nµσ√n

d→ N(0, 1);

38

i.e.,

limn→∞P

Sn − nµσ√n

< x

=1√2π

∫ x−∞e−

t2

2 dt for all x ∈ R.

Lemma 2.3.9. Let Sn =n∑i=1

Xi where Xn, n ≥ 1 is a sequence of i.i.d. random

variables with EX1 = 0, EX 21 = 1. Then

lim supn→∞

Sn√n=∞ a. c.

Proof : For arbitrary 1 ≤ M < ∞,

P

lim supn→∞

Sn√n≥ M

≥ P

Sn√n≥ M i.o.(n)

= P

∞∩n=1

∞∪k=n

[Sk√k≥ M

]

= limn→∞P

∞∪k=n

[Sk√k≥ M

]

≥ lim supn→∞

P

Sn√n≥ M

=1√2π

∫ ∞

M

e−t2

2 dt (by Lemma 2.3.8)

> 0.

By the Kolmogorov Zero-One Law (e.g., Chow and Teicher (1997, p. 64)),

P

lim supn→∞

Sn√n≥ M

= 1.

Therefore,

P

lim supn→∞

Sn√n=∞

= P

∞∩M=1

[lim supn→∞

Sn√n≥ M

]= 1. 2

Remark 2.3.3. Lemma 2.3.9 also follows immediately from the Hartman and

Wintner (1941) law of the iterated logarithm.

39

A random element V0 is said to be stochastically dominated by a random element V

if for some constant D < ∞,

P∥V0∥ > t ≤ DP∥DV ∥ > t, t ≥ 0. (2.6)

A sequence of random elements Vn, n ≥ 1 is said to be stochastically dominated by a

random element V if for some constant D < ∞,

P∥Vn∥ > t ≤ DP∥DV ∥ > t, t ≥ 0, n ≥ 1. (2.7)

Stochastic domination of Vn, n ≥ 1 is of course automatic with V = V1 and D = 1

if the random elements Vn, n ≥ 1 are identically distributed. It follows from Lemma

5.2.2 of Taylor (1978, p. 123) (or Lemma 3 of Wei and Taylor (1978)) that stochastic

dominance of a sequence of random elements Vn, n ≥ 1 can be accomplished by

the random elements in the sequence having a bounded absolute r th moment (r > 0).

Specifically, if supn≥1 E∥Vn∥r < ∞ for some r > 0, then there exists a random element V

with E∥V ∥s < ∞ for all 0 < s < r such that (2.7) holds with D = 1. (The provision that

r > 1 in Lemma 5.2.2 of Taylor (1978, p. 123) (or Lemma 3 of Wei and Taylor (1978)) is

not needed as was pointed out by Adler, Rosalsky, and Taylor (1992).)

Lemma 2.3.10 (Adler, Rosalsky, and Taylor (1989)). Let V0 and V be random

elements in a real separable Banach space such that V0 is stochastically dominated by

V in the sense that (2.6) holds for some constant D < ∞. Then

E [∥V0∥I (∥V0∥ > t)] ≤ DE [∥DV ∥I (∥DV ∥ > t)], t ≥ 0.

Lemma 2.3.11 (Adler and Rosalsky (1987a)). Let V0 and V be random elements

in a real separable Banach space such that V0 is stochastically dominated by V in the

sense that (2.6) holds for some constant D < ∞. Then for all q > 0 and t ≥ 0,

E [∥V0∥qI (∥V0∥ ≤ t)] ≤ DtqP∥DV ∥ > t+DE [∥DV ∥qI (∥DV ∥ ≤ t)].

40

Lemma 2.3.12 (Adler and Rosalsky (1987a)). Let Vn, n ≥ 1 be a sequence of

random elements in a real separable Banach space X . Suppose that Vn, n ≥ 1 is

stochastically dominated by a random element V in X in the sense that (2.7) holds for

some constant D < ∞. Let cn, n ≥ 1 be a sequence of positive constants such that(max1≤j≤n

cpj

) ∞∑j=n

1

cpj= O(n) for some p > 0

and

∞∑n=1

P∥V ∥ > Dcn < ∞. (2.8)

Then for all 0 < M < ∞,

∞∑n=1

1

cpnE [∥Vn∥pI (∥Vn∥ ≤ Mcn)] < ∞.

Finally, some remarks about notation are in order. The symbol C denotes

throughout a generic constant (0 < C < ∞) whose actual value is not important

and which is not necessarily the same one in each appearance. Furthermore, it

proves convenient to define a ∧ b = mina, b, a ∨ b = maxa, b, a, b ∈ R and

log x = loge(e ∨ x), x > 0 where loge denotes the logarithm to the base e.

41

CHAPTER 3STRONG LAWS OF LARGE NUMBERS IN RADEMACHER TYPE p (1 ≤ p ≤ 2)

BANACH SPACES FOR INDEPENDENT SUMMANDS

3.1 Objective

With the preliminaries accounted for in Chapter 2, our objective in this chapter is

to establish very general SLLNs for normed weighted sums of independent Banach

space valued random elements which are not necessarily identically distributed. The

underlying Banach space is assumed to be of Rademacher type p (1 ≤ p ≤ 2) and the

sequence of random elements is assumed to be stochastically dominated by a random

element. The main results that will be established are Theorems 3.2.1 and 3.2.2, which

are new even when the underlying Banach space is the real line. Special cases of the

main results include results of Woyczynski (1980), Teicher (1985), Adler, Rosalsky, and

Taylor (1989), and Sung (1997).

3.2 Main Results

The first main result, Theorem 3.2.1, may be presented. Its proof will be given after

Remark 3.2.1 and Example 3.2.1.

We now present the first main result, Theorem 3.2.1, which is a new result when the

underlying Banach space is the real line R. Its proof will be given after Remark 3.2.1 and

Example 3.2.1.

Theorem 3.2.1. Let 1 ≤ p ≤ 2 and let Vn, n ≥ 1 a sequence of independent

random elements in a real separable Rademacher type p Banach space X . Suppose

that Vn, n ≥ 1 is stochastically dominated by a random element V in the sense that

(2.7) holds for some constant D < ∞. Let an, n ≥ 1 and bn, n ≥ 1 be sequences of

constants satisfying 0 < bn ↑ ∞ and

anbn= O

(1

cn

)(3.1)

42

where cn, n ≥ 1 is a sequence of constants satisfying 0 < cn ↑,

cpn

∞∑j=n

1

cpj= O(n) (3.2)

and

cn

n∑j=1

1

cj= O(n). (3.3)

If

∞∑n=1

P∥V ∥ > Dcn < ∞, (3.4)

then Vn, n ≥ 1 obeys the SLLN∑ni=1 ai(Vi − EVi)

bn→ 0 a. c. (3.5)

Remark 3.2.1. The following example shows that conditions (3.2) and (3.3) are

independent in the sense that they do not imply each other.

Example 3.2.1. Let 1 ≤ p ≤ 2 and α > 0. Let cn = nα, n ≥ 1. Then, for n ≥ 1, we

have the following inequalities.

If αp > 1, then

cpn

∞∑j=n

c−pj = nαp

∞∑j=n

j−αp

≤ nαp∫ ∞

n−1x−αpdx

= nαpx1−αp

1− αp

∣∣∣∣∞x=n−1

≤ Cnαpn1−αp

= O(n).

43

If αp ≤ 1, then

cpn

∞∑j=n

c−pj = nαp

∞∑j=n

j−αp =∞ = O(n).

If 0 < α < 1, then for n ≥ 2,

cn

n∑j=1

c−1j = nα + nα

n∑j=2

j−α

≤ nα + nα∫ n1

x−αdx

= nα + nαx1−α

1− α

∣∣∣∣nx=1

= nα +n

1− α− nα

1− α= O(n).

If α = 1, then

cn

n∑j=1

c−1j = n

n∑j=1

j−1 ≥ n∫ n+11

1

xdx = n log x

∣∣∣n+1x=1= n log(n + 1)

and so

cn

n∑j=1

c−1j = O(n).

If α > 1, then

cn

n∑j=1

c−1j = nα

n∑j=1

j−α

≥ nα∫ n+11

x−αdx

= nαx1−α

1− α

∣∣∣∣n+1x=1

=nα

(1− α)(n + 1)α−1+nα

α− 1

and so

cn

n∑j=1

c−1j = O(n).

In summary, we have the following four cases:

44

(i) Both conditions (3.2) and (3.3) hold if αp > 1 and 0 < α < 1; i.e.,

1/p < α < 1 and 1 < p ≤ 2.

(ii) Both conditions (3.2) and (3.3) fail if αp ≤ 1 and α ≥ 1; i.e.,

α = p = 1.

(iii) Condition (3.2) holds but condition (3.3) fails if αp > 1 and α ≥ 1; i.e.,

α > 1 if p = 1 or α ≥ 1 if 1 < p ≤ 2.

(iv) Condition (3.2) fails but condition (3.3) holds if αp ≤ 1 and 0 < α < 1; i.e.,

0 < α < 1 if p = 1 or 0 < α ≤ 1/p if 1 < p ≤ 2.

Proof of Theorem 3.2.1.

Set c0 = 0. Note at the outset that cn ↑ ∞ by (3.2), and that cn = O(n) by (3.3).

Then, for some C > 0 and for all n ≥ 1,

E∥Vn∥CD2

≤∞∑i=0

P

∥∥∥∥ VnCD2∥∥∥∥ > i

≤

∞∑i=0

P∥Vn∥ > D2ci (since cn = O(n))

≤∞∑i=0

DP∥V ∥ > Dci (by (2.7))

< ∞ (by (3.4)),

which implies by 3. of Proposition 2.2.2 that the Vn, n ≥ 1 all have expected values.

Define

Wn = VnI (∥Vn∥ ≤ D2cn), n ≥ 1.

We shall prove the following three statements:

(i)∑∞n=1 PVn =Wn < ∞.

(ii)∑ni=1 ai(Wi − EWi)

bn→ 0 a. c.

45

(iii)∑ni=1 ai(EWi − EVi)

bn→ 0.

We prove (i) as follows. Note that

∞∑n=1

PVn =Wn =∞∑n=1

P∥Vn∥ > D2cn

≤ D∞∑n=1

P∥V ∥ > Dcn (by (2.7))

< ∞ (by (3.4)).

We prove (ii) as follows. Since Vn, n ≥ 1 and cn, n ≥ 1 satisfy the conditions of

Lemma 2.3.12,

∞∑n=1

E∥Wn∥p

cpn=

∞∑n=1

1

cpnE [∥Vn∥pI (∥Vn∥ ≤ D2cn)] < ∞. (3.6)

Thus, for n ≥ 1,

supm>nE

∥∥∥∥∥m∑i=1

ai(Wi − EWi)bi

−n∑i=1

ai(Wi − EWi)bi

∥∥∥∥∥p

= supm>nE

∥∥∥∥∥m∑

i=n+1

ai(Wi − EWi)bi

∥∥∥∥∥p

≤ C supm>n

m∑i=n+1

∣∣∣∣aibi∣∣∣∣p E∥(Wi − EWi)∥p (since X is of Rademacher type p)

≤ C2p supm>n

m∑i=n+1

E∥Wi∥p

cpi(by (3.1))

= C2p∞∑

i=n+1

E∥Wi∥p

cpi

= o(1) (by (3.6)).

Therefore, by Lemma 2.3.7,

E

∥∥∥∥∥n∑i=1

ai(Wi − EWi)bi

− S

∥∥∥∥∥p

→ 0

46

for some X -valued random element S . Thus,

n∑i=1

ai(Wi − EWi)bi

P→ S

which, by Lemma 2.3.1, implies

n∑i=1

ai(Wi − EWi)bi

→ S a. c.

Hence we obtain (ii) via Lemma 2.3.5.

We prove (iii) as follows. Note that

∞∑n=1

∥an(EWn − EVn)∥bn

≤∞∑n=1

|an|bnE [∥Vn∥I (∥Vn∥ > D2cn)]

≤ D2∞∑n=1

|an|bnE [∥V ∥I (∥V ∥ > Dcn)] (by Lemma 2.3.10)

≤ C∞∑n=1

1

cnE [∥V ∥I (∥V ∥ > Dcn)] (by (3.1))

= C

∞∑n=1

1

cn

∞∑i=n

E [∥V ∥I (Dci < ∥V ∥ ≤ Dci+1)] (since cn ↑ ∞)

≤ C∞∑i=1

E [∥V ∥I (Dci < ∥V ∥ ≤ Dci+1)]i+1∑n=1

1

cn

≤ C∞∑i=1

ci+1PDci < ∥V ∥ ≤ Dci+1C(i + 1)

ci+1(by (3.3))

≤ C∞∑i=1

iPDci < ∥V ∥ ≤ Dci+1

= C

∞∑i=1

i∑n=1

PDci < ∥V ∥ ≤ Dci+1

= C

∞∑n=1

∞∑i=n

PDci < ∥V ∥ ≤ Dci+1

= C

∞∑n=1

P∥V ∥ > Dcn < ∞ (by (3.4)).

47

Then, by Lemma 2.3.5 in the random variable case (the real line version of the

Kronecker lemma),

∥∑ni=1 aiE(Vi −Wi)∥

bn≤∑ni=1 ∥aiE(Vi −Wi)∥

bn→ 0

proving (iii).

Since (i) ensures that Plim infn→∞ [Vn = Wn] = 1 by Lemma 2.3.4, then we have

from (ii) that ∑ni=1 ai(Vi − EWi)

bn→ 0 a. c.

Combining this with (iii) yields the SLLN (3.5). 2

Remark 3.2.2. In Theorem 3.2.1, there is a trade-off between the Rademacher type

and condition (3.2); the larger p is, a more stringent condition is imposed on the Banach

space X whereas condition (3.2) becomes less stringent. To see this, suppose that (3.2)

holds for some p ∈ [1, 2) and let p′ ∈ (p, 2]. Since 0 < cn ↑,

0 <cncj

≤ 1 for j ≥ n ≥ 1,

and hence (cncj

)p′≤(cncj

)pfor j ≥ n ≥ 1

implying

cp′

n

∞∑j=n

1

cp′

j

=

∞∑j=n

(cncj

)p′≤

∞∑j=n

(cncj

)p= cpn

∞∑j=n

1

cpj= O(n).

Therefore, condition (3.2) holds for p′ ∈ (p, 2].

Remark 3.2.3. Adler and Rosalsky (1987a) showed that if cpn /n ↑, then (3.2) is

equivalent to the structurally simpler condition

lim infn→∞

cprncpn

> r for some integer r ≥ 2.

Note that strict inequality appears in this condition.

48

The first corollary of Theorem 3.2.1 is the following proposition, Proposition 3.2.1,

which is Theorem 1 of Adler, Rosalsky, and Taylor (1989) and is one of the main results

of that article.

Proposition 3.2.1 (Adler, Rosalsky, and Taylor (1989, Theorem 1)). Let 1 ≤ p ≤ 2

and let Vn, n ≥ 1 a sequence of independent random elements in a real separable

Rademacher type p Banach space X . Suppose that Vn, n ≥ 1 is stochastically

dominated by a random element V in the sense that (2.7) holds for some constant

D < ∞. Let an, n ≥ 1 and bn, n ≥ 1 be sequences of constants satisfying

0 < bn ↑ ∞, bn/|an| ↑,

bpn|an|p

∞∑j=n

|aj |p

bpj= O(n),

and

bn|an|

n∑j=1

|aj |bj= O(n). (3.7)

Moreover, suppose that∑∞n=1 P∥anV ∥ > Dbn < ∞. Then the SLLN (3.5) holds.

Proof. Take cn = bn/an, n ≥ 1 in Theorem 3.2.1. The corollary follows immediately. 2

We now present two illustrative examples, Examples 3.2.2 and 3.2.3, to show that

Theorem 3.2.1 can fail if the Rademacher type p hypothesis is dispensed with. Recall

that the real separable Banach spaces c0 and ℓ1 are not of Rademacher type p for any

1 < p ≤ 2. Example 3.2.2 presents a sequence of i.i.d. random elements in c0 which

do not satisfy the SLLN (3.5), and Example 3.2.3 presents a sequence of independent

but not identically distributed random elements in ℓ1 that do not satisfy the SLLN (3.5).

Example 3.2.3 was inspired by another one due to Beck (1963).

Example 3.2.2. In Theorem 3.2.1, let 1 < q < p < 2 and let X = c0. Let

an = 1, bn = cn = n1/q, n ≥ 1.

49

Define αn = n(1−q)/q, n ≥ 1. Then αn ↓ 0. Let ξk , k ≥ 1 be a sequence of independent

random variables with distributions given by ξ1 = ... = ξ7 = 0 a. c. and

Pξk = 1 = Pξk = −1 = 12(1− Pξk = 0) =

1

log k, k ≥ 8.

For k ≥ 1, define βk =√αn where n is such that 2n−1 ≤ k < 2n and take V to be the

random element in c0 with coordinates (βkξk)k≥1. Then V is clearly almost certainly

bounded and has zero expectation. Let Vn, n ≥ 1 be a sequence of i.i.d. copies of V .

Then conditions (3.1) and (3.4) hold. Both conditions (3.2) and (3.3) hold by Example

3.2.1 (i) taking α = 1/q. In summary, all of the conditions of Theorem 3.2.1 are satisfied

except for X being of Rademacher type p. However, in Example 7.11 of Ledoux and

Talagrand (1991, p. 190), it is shown that∑ni=1 Vi/nαn

P9 0, which implies∑ni=1(Vi − EVi)n1/q

9 0 a. c. (3.8)

Hence, the SLLN (3.5) fails.

Remark 3.2.4. Note that it follows immediately from (3.8) and q < p that∑ni=1(Vi − EVi)n1/p

9 0 a. c.

Consequently, since c0 is not of Rademacher type p, this example demonstrates

explicitly that the Marcinkiewicz-Zygmund SLLN can fail in a Banach space setting

without the Rademacher type p hypothesis. As was discussed in Chapter 1, de Acosta

(1981) proved that the validity of the Marcinkiewicz-Zygmund SLLN (1.5) in a real

separable Banach space X for every sequence of i.i.d. random elements Vn, n ≥ 1

with E∥V1∥p < ∞ for some 1 ≤ p < 2 is equivalent to the Banach space X being of

Rademacher type p. However, de Acosta (1981) did not provide an explicit example

wherein the SLLN fails for a Banach space which is not of Rademacher type p.

Example 3.2.3. In Theorem 3.2.1, let 1 < q < p < 2 and let X = ℓ1. Let

an = 1, bn = cn = n1/q, n ≥ 1.

50

For n ≥ 1, let v (n) be the element of ℓ1 having 1 in its nth position and 0 elsewhere.

Define a sequence of independent random elements Vn, n ≥ 1 in ℓ1 by requiring the

Vn, n ≥ 1 to be independent with

PVn = v (n) = PVn = −v (n) = 1/2, n ≥ 1.

Then Vn, n ≥ 1 is stochastically dominated in the sense that (2.7) holds with V = V1

and D = 1, but is not comprised of identically distributed random elements. Clearly V1

satisfies condition (3.4). As in Example 3.2.2, all of the conditions of Theorem 3.2.1 are

satisfied except for X being of Rademacher type p. Furthermore, since q > 1,

∥∑ni=1 Vi∥1n1/q

= n1−1q → ∞ = 0 a. c.

Hence, the SLLN (3.5) fails.

Remark 3.2.5. If in Example 3.2.3 we instead take the real separable Banach space

X to be ℓq (which is not of Rademacher type p), then arguing as in Example 3.2.3 we

obtain∥∑ni=1 Vi∥qn1/q

=n1/q

n1/q= 19 0 a. c.

Hence, the SLLN (3.5) fails.

The following example, Example 3.2.4, shows that Theorem 3.2.1 can fail if

condition (3.4) does not hold.

Example 3.2.4. Let Xn, n ≥ 1 be a sequence of i.i.d. random variables with

X1 ∈ L1 but X1 ∈ Lq for some q ∈ (1, 2). Let p ∈ (q, 2] and define a sequence of

independent random elements Vn, n ≥ 1 in ℓp (which is of Rademacher type p) by

Vn = (Xn, 0, 0, ...), n ≥ 1.

Then Vn, n ≥ 1 is stochastically dominated in the sense that (2.7) holds with V = V1

and D = 1. Now for each n ≥ 1, the expected value of Vn exists since E∥Vn∥p =

51

E(|Xn|p)1/p = E |Xn| < ∞. Let

an = 1, bn = cn = n1/q, n ≥ 1.

Then condition (3.1) clearly holds and conditions (3.2) and (3.3) hold by Example 3.2.1

(i). Note that E∥V1∥q = E |X1|q =∞ and so

∞∑n=1

P∥V1∥ > n1/q =∞;

that is, condition (3.4) fails. Thus, all of the hypotheses of Theorem 3.2.1 are satisfied

except for (3.4). Note that for all n ≥ 1,∑ni=1(Vi − EVi)n1/q

=(∑ni=1(Xi − EXi), 0, 0, ...)

n1/q

and so ∥∥∥∥∑ni=1(Vi − EVi)n1/q

∥∥∥∥p

=(|∑ni=1(Xi − EXi)|p)1/p

n1/q

=|∑ni=1(Xi − EXi)|n1/q

9 0 a. c.

by the real line version of the Marcinkiewicz-Zygmund SLLN recalling that E |X1|q = ∞.

Hence, the SLLN (3.5) fails.

The following example, Example 3.2.5, shows that the hypotheses of Theorem 3.2.1

are satisfied but those of Proposition 3.2.1 (Theorem 1 of Adler, Rosalsky, and Taylor

(1989)) are not satisfied. Consequently, Theorem 3.2.1 is a bona fide improvement of

Proposition 3.2.1 (Theorem 1 of Adler, Rosalsky, and Taylor (1989)).

Example 3.2.5. Let 1 < p ≤ 2 and γ ≥ 1. Suppose that

bn/|an| = nγ and cn = nα where 1/p < α < 1.

Let Vn, n ≥ 1 be a sequence of independent random elements in a real separable

Rademacher type p Banach space which are stochastically dominated by a random

52

element V in the sense that (2.7) holds and E∥V ∥1/α < ∞. Then,

∞∑n=1

P∥V ∥ > Dcn =∞∑n=1

P

∥∥∥∥VD∥∥∥∥ > cn

=

∞∑n=1

P

∥∥∥∥VD∥∥∥∥ > nα

=

∞∑n=1

P

∥∥∥∥VD∥∥∥∥1/α > n

≤ E∥∥∥∥VD

∥∥∥∥1/α < ∞.

Thus, condition (3.4) holds. Since α− γ < 0, it follows that

cn|an|bn= nα−γ = o(1),

which implies that condition (3.1) holds. Furthermore, by Example 3.2.1 (i), both

conditions (3.2) and (3.3) hold. In summary, all of the conditions of Theorem 3.2.1 are

satisfied.

On the other hand, if γ > 1, then

bn|an|

n∑j=1

|aj |bj= nγ

n∑j=1

j−γ = O(n).

whereas if γ = 1, then

bn|an|

n∑j=1

|aj |bj= n

n∑j=1

j−1 = (1 + o(1))n log n = O(n).

Therefore, condition (3.7) of Proposition 3.2.1 fails. Consequently, Theorem 3.2.1

ensures that Vn, n ≥ 1 obeys the SLLN (3.5) whereas Proposition 3.2.1 (Theorem 1 of

Adler, Rosalsky, and Taylor (1989)) is not applicable for this example.

The second corollary of Theorem 3.2.1 is the following theorem, Theorem 3.2.2,

which is an improved version of Theorem 6 of Adler, Rosalsky, and Taylor (1989) as will

be discussed in detail in Remark 3.2.6 below. Moreover, Theorem 3.2.2 is a new result

when the underlying Banach space is the real line R.

53

Theorem 3.2.2. Let 1 < p ≤ 2 and let Vn, n ≥ 1 a sequence of independent

random elements in a real separable Rademacher type p Banach space X . Suppose

that Vn, n ≥ 1 is stochastically dominated by a random element V in the sense that

(2.7) holds for some constant D < ∞. Moreover, suppose that E∥V ∥q < ∞ for some

1 < q < p. Let an, n ≥ 1 and bn, n ≥ 1 be sequences of constants satisfying

0 < bn ↑ ∞ and

anbn= O

(1

n1/q

). (3.9)

Then the SLLN (3.5) holds.

Proof. Take cn = n1/q, n ≥ 1 in Theorem 3.2.1. Then (3.9) is precisely (3.1). Note that

E∥V ∥q < ∞ implies (3.4). Moreover, (3.2) holds since p > q, and (3.3) holds since

q > 1. The conclusion (3.5) follows immediately from Theorem 3.2.1. 2

The following proposition, Proposition 3.2.2, which is Theorem 2 of Sung (1997) and

is one of the main results of that article, is a direct corollary of Theorem 3.2.2.

Proposition 3.2.2 (Sung (1997, Theorem 2)). Let 1 < p < 2 and let Xn, n ≥ 1 be

a sequence of i.i.d. Lp random variables. Let an, n ≥ 1 and bn, n ≥ 1 be sequences

of constants with 0 < bn ↑ ∞ and

anbn= O

(1

n1/p

).

Then the SLLN ∑ni=1 ai(Xi − EXi)

bn→ 0 a. c.

holds.

Proof. Recall that the real line is of Rademacher type 2. In Theorem 3.2.2, take X to be

the real line, take p = 2, and take q to be the p in the statement of this proposition. The

proposition follows immediately. 2

54

Remark 3.2.6. (i) Theorem 3.2.2 was proved by Adler, Rosalsky, and Taylor(1989, Theorem 6) using the additional condition

n∑i=1

|ai | = O(bn). (3.10)

Theorem 3.2.2 is also valid for q = 1 provided condition (3.10) holds as was provedby Adler, Rosalsky, and Taylor (1989) in their Theorem 6. The following example ofSung (1997) shows that condition (3.10) cannot be dispensed with when q = 1:

Let the underlying real separable Banach space X be R which is of Rademachertype p = 2. Let Vn, n ≥ 1 be a sequence of i.i.d. random variables with V1 havingprobability density function

f (x) =c

x2(log x)2I[2,∞)(x), −∞ < x < ∞

where c is a positive constant. Then EV1 = c/ log 2. Let an = 1/n, 0 < bn ↑ ∞ andbn = o(log(log n)) for all n ≥ 1. Then (3.9) holds with q = 1, (3.10) fails, and∑n

i=1 ai(Vi − EVi)bn

→ −∞ a. c.

Thus (3.5) fails.

(ii) Theorem 6 of Adler, Rosalsky, and Taylor (1989) extends Theorem 2 of Adlerand Rosalsky (1987b) from the real line to a Rademacher type p (1 < p ≤ 2)Banach space. Theorem 2 of Sung (1997), which does not have (3.10) as anassumption, thus improves Theorem 2 of Adler and Rosalsky (1987b).

(iii) Note that the larger q is in Theorem 3.2.2, condition E∥V ∥q < ∞ is strongerwhereas condition (3.9) is weaker.

(iv) In the special case where Vn, n ≥ 1 is a sequence of i.i.d. random elementswith E∥V1∥q < ∞, an = 1, bn = n1/q, n ≥ 1, Theorem 3.2.2 reduces to theMarcinkiewicz-Zygmund type SLLN∑n

i=1(Vi − EVi)n1/q

→ 0 a. c. (3.11)

of Woyczynski (1980), Theorem 4.1. This result of Woyczynski (1980) of coursedoes not follow from Theorem 6 of Adler, Rosalsky, and Taylor (1989) becausecondition (3.10) fails with an, n ≥ 1 and bn, n ≥ 1 as above. A strongerresult was obtained by de Acosta (1981, Theorem 4.1) and by Azlarov and Volodin(1981) who showed that (3.11) holds under the assumption that the Banach spaceX is of Rademacher type q (which is weaker than X being Rademacher type p).

55

(v) Example 3.2.2 also demonstrates that Proposition 3.2.1, Theorem 3.2.2, andthe Marcinkiewicz-Zygmund type SLLN of Woyczynski (1980) ((iv) above) can fail ifthe Rademacher type p (1 < p ≤ 2) hypothesis is dispensed with.

The following example, Examples 3.2.6, shows that the conditions of Theorem 3.2.1

are satisfied but the conditions of Theorem 3.2.2 are not satisfied.

Example 3.2.6. Let the underlying real separable Banach space X be R which is

of Rademacher type p = 2. Let 3/2 < α < 2 and let Vn, n ≥ 1 be a sequence of i.i.d.

random variables with V1 having probability density function

f (x) =c

|x |α

α−1 (log |x |)3I[e,∞)(|x |), −∞ < x < ∞

where c is a positive constant. Note at the outset that EV1 = 0 noting V1 has a

symmetric distribution and α/(α− 1) > 2 since 3/2 < α < 2. Let

an = log n, bn = nα−1, and cn =

nα−1

log(n + 1), n ≥ 1.

Then condition (3.1) and 0 < cn ↑ hold. Now we prove that conditions (3.2) and (3.3)

hold as follows. Let Z(x) = (log(x + 1))2 and u = −2(α − 1). Then Z is slowly varying;

that is, Z varies regularly with exponent ρ = 0. Define Z ∗u (x) as in (2.5). By Lemma

2.3.2, Z ∗u (x) < ∞ since α > 3/2. Then, by Lemma 2.3.3 (i),

limx→∞

xu+1Z(x)

Z ∗u (x)

= −(u + ρ+ 1) = −u − 1 = 2α− 3 > 0.

Thus, for n ≥ 2,

cpn

∞∑j=n

c−pj =n2(α−1)

(log(n + 1))2

∞∑j=n

(log(j + 1))2

j2(α−1)

≤ n2(α−1)

(log(n + 1))2

∫ ∞

n−1

(log(y + 1))2

y 2(α−1)dy

=Z ∗u (n − 1)nuZ(n)

= (1 + o(1))(n − 1)u+1Z(n − 1)(2α− 3)nuZ(n)

= O(n)

56

thereby proving (3.2).

Similarly, let Z(x) = log x and u = 1 − α. Then Z is slowly varying; that is, Z varies

regularly with exponent ρ = 0. Define Zu(x) as in (2.5). Now α < 2 ensures that u > −1

and so by Lemma 2.3.3 (ii),

limx→∞

xu+1Z(x)

Zu(x)= u + ρ+ 1 = u + 1 = 2− α > 0.

Thus,

cn

n∑j=1

c−1j =nα−1

log(n + 1)

n∑j=1

log(j + 1)

jα−1

≤ nα−1

log(n + 1)

∫ n0

log(y + 1)

yα−1dy

=Zu(n)

nuZ(n + 1)

= (1 + o(1))nu+1Z(n)

(2− α)nuZ(n + 1)

= O(n)

thereby proving (3.3).

Furthermore, we prove that condition (3.4) holds with D = 1 as follows. Let

Z(x) = (log x)−3 and u = −α/(α − 1). Then Z is slowly varying; that is, Z varies

regularly with exponent ρ = 0. Define Z ∗u (x) as in (2.5). By Lemma 2.3.2, Z ∗

u (x) < ∞

since u < −1. Then, by Lemma 2.3.3 (i),

limx→∞

xu+1Z(x)

Z ∗u (x)

= −(u + ρ+ 1) = −u − 1 = 1

α− 1> 0.

Thus,

P|V1| > x = 2∫ ∞

x

f (y)dy = 2c

∫ ∞

x

y uZ(y)dy

= 2cZ ∗u (x) = (1 + o(1))2c(α− 1)xu+1Z(x)

57

which implies

P|V1| > x = (1 + o(1))2c(α− 1)x

1α−1 (log x)3

as x → ∞. (3.12)

Then by (3.12),

P|V1| > cn = (1 + o(1))2c(α− 1)(log(n + 1))

1α−1

n[(α− 1) log n − log log(n + 1)]3

= (1 + o(1))2c(α− 1)(log n)

1α−1

n(α− 1)3(log n)3

= (1 + o(1))2c

(α− 1)2n(log n)3α−4α−1

and so condition (3.4) holds with V = V1 and D = 1 since α > 3/2 (hence (3α− 4)/(α−

1) > 1). Then, by Theorem 3.2.1, the SLLN∑ni=1(log i)Xi

nα−1→ 0 a. c.

holds.

To show that the hypotheses of Theorem 3.2.2 are not satisfied, note that condition

(3.9) holds for some q ∈ (1, 2) if and only if 1/q < α − 1; that is, q > (α − 1)−1. On the

other hand, by (3.12),

∞∑n=1

P|V1| > n1/q =∞∑n=1

(1 + o(1))2c(α− 1)q3

n1

q(α−1) (log n)3< ∞

if and only if (q(α − 1))−1 ≥ 1; that is, q ≤ (α − 1)−1. Therefore, in Theorem 3.2.2,

condition (3.9) and the condition that E |V |q < ∞ for some 1 < q < p cannot both hold.

Thus Theorem 3.2.2 is not applicable for this example.

The following example, Example 3.2.7, shows that the conditions of Theorem 3.2.2

are satisfied but those of Proposition 3.2.3 (Theorem 3 of Adler, Rosalsky, and Taylor

(1989)) are not satisfied.

We first give the statement of Theorem 3 of Adler, Rosalsky, and Taylor (1989).

58

Proposition 3.2.3 (Adler, Rosalsky, and Taylor (1989, Theorem 3)). Let 1 ≤ p ≤ 2

and let Vn, n ≥ 1 a sequence of independent random elements in a real separable

Rademacher type p Banach space X . Suppose that Vn, n ≥ 1 is stochastically

dominated by a random element V in the sense that (2.7) holds for some constant

D < ∞, and that

P∥V ∥ > x is regularly varying with exponent ρ < −1.

Let an, n ≥ 1 and bn, n ≥ 1 be sequences of constants satisfying 0 < bn ↑ ∞ and(max1≤j≤n

bpj|aj |p

) ∞∑j=n

|aj |p

bpj= O(n). (3.13)

If

∞∑n=1

P∥anV ∥ > Dbn < ∞, (3.14)

then the SLLN (3.5) holds.

Example 3.2.7. Let 1 < p ≤ 2 and let Vn, n ≥ 1 a sequence of independent

random elements in a real separable Rademacher type p Banach space X . Suppose

that Vn, n ≥ 1 is stochastically dominated by a random element V in the sense that

(2.7) holds for some constant D < ∞. Moreover, suppose that E∥V ∥q < ∞ for some

1 < q < p. Let an, n ≥ 1 and bn, n ≥ 1 be sequences of constants satisfying

0 < bn ↑ ∞ and

an =

bn2n/p

for n odd,

bn4n/p

for n even.

Then (3.9) holds sinceanbn

≤ 1

2n/p= O

(1

n1/q

).

59

In summary, all of the conditions of Theorem 3.2.2 are satisfied. However, for n even,

max1≤j≤n+1

bj|aj |

n + 1

∞∑j=n+1

|aj |bj

≥ 4n/p

n + 1· 1

2(n+1)/p

=2n−1p

n + 1

→ ∞ as n even approaches ∞.

Therefore, condition (3.13) of Proposition 3.2.3 (Theorem 3 of Adler, Rosalsky, and

Taylor (1989)) fails.

The following example, Example 3.2.8, shows that the conditions of Proposition

3.2.3 (Theorem 3 of Adler, Rosalsky, and Taylor (1989)) are satisfied but those of

Theorem 3.2.2 are not satisfied.

Example 3.2.8. Let the underlying real separable Banach space X be R which is

of Rademacher type p = 2 and let Vn, n ≥ 1 be a sequence of i.i.d. random variables

with V1 having probability density function

f (x) =c

|x |1+α(log |x |)3I[e,∞)(|x |), −∞ < x < ∞

where 1 < α < 2 and c is a positive constant. Note at the outset that V1 is integrable

and hence EV1 = 0 since V1 has a symmetric distribution. Let Z(x) = (log x)−3 and

u = −(1 + α). Then Z is slowly varying; that is, Z varies regularly with exponent

0. Define Z ∗u (x) as in (2.5). By Lemma 2.3.2 Z ∗

u (x) varies regularly with exponent

ρ = u + 1 = −α. Note that

P|V1| > x = 2∫ ∞

x

f (y)dy = 2c

∫ ∞

x

y uZ(y)dy = 2cZ ∗u (x). (3.15)

Therefore, P|V1| > x is regularly varying with exponent ρ = −α < −1. Now let

an = log(n + 1), bn = n1/α, n ≥ 1.

60

We prove that condition (3.14) holds with D = 1 as follows. By Lemma 2.3.3 (i),

limx→∞

xu+1Z(x)

Z ∗u (x)

= −(u + 1) = α > 0.

Then by (3.15),

P|V1| > x = 2cZ ∗u (x) = (1 + o(1))2c

xu+1Z(x)

−u − 1

= (1 + o(1))2c

αxα(log x)3

(3.16)

and so

P|anV1| > Dbn = P|V1| >

n1/α

log(n + 1)

= (1 + o(1))

2c(log(n + 1))α

αn[ 1αlog n − log log(n + 1)]3

= (1 + o(1))2c(log n)α

αnα−3(log n)3

= (1 + o(1))2cα2

n(log n)3−α.

Thus condition (3.14) holds since 3− α > 1.

Furthermore, we prove that condition (3.13) holds as follows. Let Z(x) = (log(x +

1))2 and u = −2/α. Then Z is slowly varying; that is, Z varies regularly with exponent

ρ = 0. Define Z ∗u (x) as in (2.5). By Lemma 2.3.2, Z ∗

u (x) < ∞ since α < 2. Then, by

Lemma 2.3.3 (i),

limx→∞

xu+1Z(x)

Z ∗u (x)

= −(u + ρ+ 1) = −u − 1 = 2− α

α> 0.

Thus, for n ≥ 2, (max1≤j≤n

bpj|aj |p

) ∞∑j=n

|aj |p

bpj

=n2/α

(log(n + 1))2

∞∑j=n

(log(j + 1))2

j2/α

≤ n2/α

(log(n + 1))2

∫ ∞

n−1

(log(y + 1))2

y 2/αdy

61

=Z ∗u (n − 1)nuZ(n)

=(1 + o(1))α(2− α)−1(n − 1)u+1Z(n − 1)

nuZ(n)

=O(n)

thereby proving (3.13). In summary, all of the conditions of Proposition 3.2.3 (Theorem 3

of Adler, Rosalsky, and Taylor (1989)) are satisfied.

To show that the hypotheses of Theorem 3.2.2 are not satisfied, note that condition

(3.9) holds for some q ∈ (1, 2) if and only if q > α. On the other hand, by (3.16),

∞∑n=1

P|V1| > n1/q =∞∑n=1

(1 + o(1))2cq3

αnα/q(log(n + 1))3< ∞

if and only if q ≤ α. Therefore, in Theorem 3.2.2, condition (3.9) and the condition that

E |V |q < ∞ for some 1 < q < p cannot both hold. Thus Theorem 3.2.2 is not applicable

for this example.

The following example, Example 3.2.9, shows that the conditions of both Theorem

3.2.2 and Proposition 3.2.3 (Theorem 3 of Adler, Rosalsky, and Taylor (1989)) are

satisfied. Consequently, Examples 3.2.7, 3.2.8, and 3.2.9 demonstrate that the

conditions of Theorem 3.2.2 and Proposition 3.2.3 (Theorem 3 of Adler, Rosalsky,

and Taylor (1989)) are compatible with each other, but that neither set of conditions

implies the other.

Example 3.2.9. Let the underlying real separable Banach space X be R which is

of Rademacher type p = 2 and let Vn, n ≥ 1 be a sequence of i.i.d. random variables

with V1 having probability density function

f (x) =c

|x |1+q(log |x |)3I[e,∞)(|x |), −∞ < x < ∞

62

where 1 < q < 2 and c is a positive constant. The same argument as in Example 3.2.8

shows that P|V1| > x is regularly varying with exponent ρ < −1. Let

an = 1, bn = n1/q, n ≥ 1.

Then condition (3.13) clearly holds by Example 3.2.1 (i) taking α = 1/q and p = 2.

Furthermore, by (3.16),

P|anV1| > bn = P|V1| > n1/q = (1 + o(1))2cq2

n(log n)3

implying condition (3.14) holds with D = 1. In summary, all of the conditions of

Proposition 3.2.3 (Theorem 3 of Adler, Rosalsky, and Taylor (1989)) are satisfied.

Clearly, condition (3.9) holds and E |V1|q < ∞. Hence, all of the conditions of Theorem

3.2.2 also hold. Consequently, Theorem 3.2.2 and Proposition 3.2.3 (Theorem 3 of

Adler, Rosalsky, and Taylor (1989)) can each be employed to establish the SLLN (3.5).

The following two examples, Examples 3.2.10 and 3.2.11, show that Theorem 3.2.2

is sharp in the sense that it can fail if condition (3.9) is weakened to

anbn= O

(1

n1/p

). (3.17)

In the first example, Example 3.2.10, the real separable Banach space is ℓp (1 < p ≤ 2)

whereas in the second example, Example 3.2.11, the real separable Banach space is R.

Example 3.2.10. Let 1 < p ≤ 2 and let the underlying real separable Banach space

X be ℓp. Then X is of Rademacher type p. Let

an = 1, bn = n1/p, n ≥ 1.

For n ≥ 1, let v (n) be the element of ℓp having 1 in its nth position and 0 elsewhere.

Define a sequence of independent random elements Vn, n ≥ 1 in ℓp by requiring the

Vn, n ≥ 1 to be independent with

PVn = v (n) = PVn = −v (n) = 1/2, n ≥ 1.

63

Then Vn, n ≥ 1 is stochastically dominated in the sense that (2.7) holds with V = V1

and D = 1. Clearly, we have E∥V1∥qp = 1 < ∞ for each q ∈ (1, p). Moreover, condition

(3.17) holds but the stronger condition (3.9) fails for all q ∈ (1, p). However,

∥∑ni=1 Vi∥pn1/p

=n1/p

n1/p= 19 0 a. c. (3.18)

Hence, the SLLN (3.5) fails. On the other hand, it follows immediately from (3.18) that for

all 0 < q < p ∑ni=1 Vi

n1/q→ 0 a. c. (3.19)

The SLLN (3.19) also follows from Theorem 3.2.2 for all 1 < q < p and, a fortiori, for all

0 < q < p.

Example 3.2.11. Let the underlying real separable Banach space be R which is of

Rademacher type p = 2. Let Vn, n ≥ 1 be a sequence of i.i.d. random variables with

Var(V1) = 1. Let

an = 1 and bn =√n, n ≥ 1.

Then (3.17) holds but the stronger condition (3.9) fails for all q ∈ (1, 2). Now by Lemma

2.3.9,

lim supn→∞

∑ni=1 ai(Vi − EVi)

bn= lim sup

n→∞

∑ni=1(Vi − EVi)√

n=∞ a. c.

Thus the SLLN (3.5) fails. It should be noted that it does follow from Theorem 3.2.2 that∑ni=1(Vi − EVi)n1/q

→ 0 a. c. (3.20)

for every q ∈ (1, 2) and, a fortiori, (3.20) holds for every q ∈ (0, 2). This is the classical

Marcinkiewicz-Zygmund SLLN.

Remark 3.2.7. We note that when condition (3.9) is dispensed with, the SLLN (3.5)

fails in different ways in Examples 3.2.10 and 3.2.11. In Example 3.2.10,

limn→∞

∥∑ni=1 ai(Vi − EVi)∥

bn= 1 a. c.

64

whereas in Example 3.2.11

lim supn→∞

|∑ni=1 ai(Vi − EVi)|

bn=∞ a. c.

The next corollary, Corollary 3.2.1, extends both Theorem 5 of Teicher (1985) and

Corollary 2 of Sung (1997). The argument is patterned after that of Corollary 2 of Sung

(1997).

Corollary 3.2.1. Let Vn, n ≥ 1 be a sequence of independent random elements

taking values in a real separable Rademacher type p (1 < p ≤ 2) Banach space and

suppose that Vn, n ≥ 1 is stochastically dominated by a random element V in the

sense that (2.7) holds for some constant D < ∞. Let an, n ≥ 1 and dn, n ≥ 1 be real

sequences where 0 < dn ↑. Suppose that E∥V ∥q < ∞,∑∞n=1 |an|q =∞, and

an(∑ni=1 |ai |q

)1/q = O ( dnn1/q)

(3.21)

for some q ∈ [1, p). Then the SLLN∑ni=1 ai(Vi − EVi)dn(∑n

i=1 |ai |q)1/q → 0 a. c. (3.22)

obtains.

Proof. Let bn = dn

(n∑i=1

|ai |q)1/q

, n ≥ 1. Then bn ↑ ∞ and by (3.21)

anbn= O

(1

n1/q

).

Now if q = 1, thenn∑i=1

|ai | =bndn

≤ bnd1= O(bn)

and the conclusion (3.22) follows from Proposition 1.3.3 (Theorem 6 of Adler, Rosalsky,

and Taylor (1989)). And if q > 1, the conclusion (3.22) follows from Theorem 3.2.2. 2

The following example illustrates Corollary 3.2.1.

65

Example 3.2.12. Let Vn, n ≥ 1 be a sequence of independent random elements

taking values in a real separable Rademacher type p (1 < p ≤ 2) Banach space and

suppose that Vn, n ≥ 1 is stochastically dominated by a random element V in the

sense that (2.7) holds for some constant D < ∞. Suppose that E∥V ∥q < ∞ for some

q ∈ [1, p). Then

(i) ∑ni=1(i log i)

−1/q(Vi − EVi)(log log n)1/q

→ 0 a. c. (3.23)

(ii) For β > −q−1, ∑ni=1 i

−1/q(log i)β(Vi − EVi)(log n)β+q−1

→ 0 a. c. (3.24)

(iii) For −∞ < α < q−1 and −∞ < β < ∞,∑ni=1 i

−α(log i)β(Vi − EVi)(log n)βnq−1−α

→ 0 a. c. (3.25)

Proof. Let dn = 1, n ≥ 1.

(i) Let an = (n log n)−1/q, n ≥ 1. Then

n∑i=1

|ai |q =n∑i=1

(i log i)−1 ∼ log log n → ∞

and (3.21) holds since

|an|q∑ni=1 |ai |q

=1

(n log n)(∑ni=1(i log i)

−1)= O

(1

n

).

The conclusion (3.23) follows from Corollary 3.2.1.

(ii) Let an = n−1/q(log n)β, n ≥ 1. Then

n∑i=1

|ai |q =n∑i=1

i−1(log i)qβ ∼ (log n)qβ+1

qβ + 1→ ∞

and (3.21) holds since

|an|q∑ni=1 |ai |q

∼ (log n)qβ(qβ + 1)

n(log n)qβ+1= O

(1

n

).

66

The conclusion (3.24) follows from Corollary 3.2.1.

(iii) Let an = n−α(log n)β, n ≥ 1. Then

n∑i=1

|ai |q =n∑i=1

i−qα(log i)qβ ∼ (log n)qβ n1−qα

1− qα

by Lemma 2.3.3 (ii). Then (3.21) holds since

|an|q∑ni=1 |ai |q

∼ n−qα(log n)qβ(1− qα)(log n)qβn1−qα

= O

(1

n

).

The conclusion (3.25) follows from Corollary 3.2.1. 2

67

CHAPTER 4STRONG LAWS OF LARGE NUMBERS FOR RANDOM ELEMENTS IN GENERAL

BANACH SPACES IRRESPECTIVE OF THEIR JOINT DISTRIBUTIONS

4.1 Objective

Our objective in this chapter is to obtain SLLNs irrespective of the joint distributions

of the random elements where in addition no geometric conditions are imposed on the

underlying Banach space. We will establish four theorems (Theorems 4.2.1, 4.2.2, 4.2.3,

and 4.2.4) all of which have the assumption that the sequence of random elements is

stochastically dominated by a random element. Theorem 4.2.1 is a “0 < q < 1” version

of Theorem 3.2.2. Theorem 4.2.2 pertains to 0 < q ≤ 1 and the 0 < q < 1 part follows

from Theorem 4.2.1. Theorems 4.2.3 and 4.2.4 are obtained separately. Theorems

4.2.1, 4.2.2, 4.2.3, and 4.2.4 are new even when the underlying Banach space is the

real line. The results are general enough to include results of Petrov (1973), Teicher

(1985), Sung (1997), and Rosalsky and Stoica (2010).

4.2 Main Results

The most general result in the literature that we are aware of establishing a SLLN

for normed weighted sums of stochastically dominated random elements Vn, n ≥ 1

irrespective of their joint distributions is the following proposition, Proposition 4.2.1,

which is Theorem 11 of Adler, Rosalsky, and Taylor (1989).

Proposition 4.2.1 (Adler, Rosalsky, and Taylor (1989, Theorem 11)). Let Vn, n ≥

1 be a sequence of random elements in a real separable Banach space X . Suppose

that Vn, n ≥ 1 is stochastically dominated by a random element V in the sense that

(2.7) holds for some constant D < ∞. Let an, n ≥ 1 and bn, n ≥ 1 be sequences of

constants satisfying 0 < bn ↑ ∞ and(max1≤j≤n

bj|aj |

) ∞∑j=n

|aj |bj= O(n). (4.1)

68

If

∞∑n=1

P∥anV ∥ > Dbn < ∞, (4.2)

then the SLLN ∑ni=1 aiVibn

→ 0 a. c. (4.3)

holds irrespective of the joint distributions of the Vn, n ≥ 1.

The first main result of Chapter 4, is Theorem 4.2.1, which is a “0 < q < 1”

version of Theorem 3.2.2. The random elements Vn, n ≥ 1 are not assumed to be

independent and the underlying Banach space is not assumed to be of Rademacher

type p for some 1 < p ≤ 2. We note that there is a trade-off between the moment

condition E∥V ∥q < ∞ and condition (4.4); the closer q is to 1, the moment condition

E∥V ∥q < ∞ becomes stronger whereas condition (4.4) becomes weaker. Theorem

4.2.1 is, in effect, a result for sums of nonnegative random variables since its statement

and proof involves the random elements only through their norms. Moreover, Theorem

4.2.1 is a new result even when the underlying Banach space is the real line R and it is

also new when an ≡ 1.

Theorem 4.2.1. Let Vn, n ≥ 1 be a sequence of random elements in a real

separable Banach space X . Suppose that Vn, n ≥ 1 is stochastically dominated by a

random element V in the sense that (2.7) holds for some constant D < ∞. Moreover,

suppose that E∥V ∥q < ∞ for some 0 < q < 1. Let an, n ≥ 1 and bn, n ≥ 1 be

sequences of constants satisfying 0 < bn ↑ ∞ and

anbn= O

(1

n1/q

). (4.4)

Then the SLLN (4.3) holds irrespective of the joint distributions of the Vn, n ≥ 1.

Proof. Define

Wn = VnI (∥Vn∥ ≤ D2n1/q), n ≥ 1.

69

In Lemma 2.3.12, let cn = n1/q for n ≥ 1 and let p = 1. Note that cpn /n = n1/q−1 ↑ and

lim infn→∞

cp2ncpn= lim inf

n→∞

(2n)1/q

n1/q= 21/q > 2

since 1/q > 1. Then (max1≤j≤n

cpj

) ∞∑j=n

1

cpj= O(n)

by Remark 3.2.3 taking r = 2. Moreover, condition (2.8) holds since E∥V ∥q < ∞. Thus,

by Lemma 2.3.12,∞∑n=1

1

cnE [∥Vn∥I (∥Vn∥ ≤ D2cn)] < ∞.

Therefore,

E

(∞∑n=1

∥anWn∥bn

)

=

∞∑n=1

E

(∥anWn∥bn

)(by Lemma 2.3.6)

≤C∞∑n=1

1

n1/qE [∥Vn∥I (∥Vn∥ ≤ D2n1/q)] (by (4.4))

=C

∞∑n=1

1

cnE [∥Vn∥I (∥Vn∥ ≤ D2cn)]

<∞

whence∞∑n=1

∥anWn∥bn

< ∞ a. c.

Then by Lemma 2.3.5 in the random variable case (the real line version of the Kronecker

lemma),∥∑ni=1 aiWi∥bn

≤∑ni=1 ∥aiWi∥bn

→ 0 a. c.

and so ∑ni=1 aiWi

bn→ 0 a. c. (4.5)

70

Note that

∞∑n=1

PVn =Wn =∞∑n=1

P∥Vn∥ > D2n1/q

≤ D∞∑n=1

P∥V ∥ > Dn1/q (by (2.7))

< ∞ (since E∥V ∥q < ∞).

Then Plim infn→∞ [Vn = Wn] = 1 by Lemma 2.3.4. Hence, in view of (4.5), we obtain

the SLLN (4.3). 2

The following example, Example 4.2.1, shows that Theorem 4.2.1 is sharp in the

sense that it can fail if condition (4.4) is weakened to

anbn= O

(1

n1/p

)for some p > q. (4.6)

Incidentally, Example 4.2.1 also has the same moral as Example 3.2.11 which

concerned a sequence of independent random variables.

Example 4.2.1. Let 0 < q < 1 ≤ p ≤ 2 and let the underlying real separable Banach

space X be ℓp. Set

an = 1, bn = n1/p, n ≥ 1.

Then (4.6) holds but the stronger condition (4.4) fails. For n ≥ 1, let v (n) be the

element of ℓp having 1 in its nth position and 0 elsewhere. Define a sequence of random

elements Vn, n ≥ 1 in ℓp by requiring

PVn = v (n) = PVn = −v (n) = 1/2, n ≥ 1.

Then Vn, n ≥ 1 is stochastically dominated by V = V1 with D = 1. Clearly, we have

E∥V1∥ qp = 1 < ∞. However,

∥∑ni=1 Vi∥pn1/p

=n1/p

n1/p= 19 0 a. c. (4.7)

Hence, the SLLN (4.3) fails.

71

The following corollary, Corollary 4.2.1, is a direct corollary of Theorem 4.2.1 and

is a new version of Theorem 5 of Teicher (1985) and Corollary 2 of Sung (1997) which

pertained to sequences of i.i.d. Lp random variables where 1 ≤ p < 2.

Corollary 4.2.1. Let Vn, n ≥ 1 be a sequence of random elements in a real

separable Banach space. Suppose that Vn, n ≥ 1 is stochastically dominated by a

random element V in the sense that (2.7) holds for some constant D < ∞ and that

E∥V ∥q < ∞ for some 0 < q < 1. Let an, n ≥ 1 and dn, n ≥ 1 be sequences of

constants satisfying∑∞n=1 |an|q =∞, 0 < dn ↑, and

an(∑ni=1 |ai |q

)1/q = O ( dnn1/q). (4.8)

Then the SLLNn∑i=1

aiVi

dn

(n∑i=1

|ai |q)1/q → 0 a. c. (4.9)

holds irrespective of the joint distributions of the Vn, n ≥ 1.

Proof. Let bn = dn

(n∑i=1

|ai |q)1/q

, n ≥ 1. Then bn ↑ ∞, and by (4.8),

anbn= O

(1

n1/q

).

Then the conclusion (4.9) follows immediately from Theorem 4.2.1. 2

The following example, Example 4.2.2, illustrates Corollary 4.2.1.

Example 4.2.2. Let Vn, n ≥ 1 be a sequence of random elements in a real

separable Banach space. Suppose that Vn, n ≥ 1 is stochastically dominated by a

random element V in the sense that (2.7) holds for some constant D < ∞ and suppose

that E∥V ∥q < ∞ for some 0 < q < 1. Then

(i) ∑ni=1(i log i)

−1/qVi

(log log n)1/q→ 0 a. c.

72

(ii) For β > −1/q, ∑ni=1 i

−1/q(log i)βVi

(log n)1/q+β→ 0 a. c.

(iii) For −∞ < α < 1/q and −∞ < β < ∞,∑ni=1 i

−α(log i)βVi(log n)βn1/q−α

→ 0 a. c.

Proof.

(i) In Corollary 4.2.1, let dn ≡ 1 and an = (n log n)−1/q, n ≥ 1. Then

∞∑n=1

|an|q =∞∑n=1

1

n log n=∞

and

an(∑ni=1 |ai |q

)1/q = (n log n)−1/q

(∑ni=1(i log i)

−1)1/q

∼ (n log n)−1/q

(log log n)1/q

= O

(dnn1/q

).

Thus the conclusion follows immediately from Corollary 4.2.1.

(ii) In Corollary 4.2.1, let dn ≡ 1 and an = n−1/q(log n)β, n ≥ 1. Then

∞∑n=1

|an|q =∞∑n=1

(log n)qβ

n=∞

since qβ > −1. Moreover,

an(∑ni=1 |ai |q

)1/q = n−1/q(log n)β(∑ni=1 i

−1(log i)qβ)1/q

∼ n−1/q(log n)β(qβ + 1)1/q

(log n)1/q+β

= O

(dnn1/q

).

Thus the conclusion again follows immediately from Corollary 4.2.1.

73

(iii) In Corollary 4.2.1, let dn ≡ 1 and an = n−α(log n)β, n ≥ 1. Then

∞∑n=1

|an|q =∞∑n=1

(log n)qβ

nqα=∞

since qα < 1. Note that

n∑i=1

|ai |q =n∑i=1

(log i)qβ

iqα∼ (log n)

qβn1−qα

1− qα

by Lemma 2.3.3 (ii). Thus,

an(∑ni=1 |ai |q

)1/q ∼ n−α(log n)β(1− qα)1/q

(log n)βn1/q−α= O

(dnn1/q

).

The conclusion again follows immediately from Corollary 4.2.1.

Remark 4.2.1. We will show below that if condition (4.4) is strengthened to the

conditionbn

|an|n1/qis quasi-monotone increasing ;

that is,

bn|an|n1/q

≤ C bj|aj |j1/q

< ∞ for some C ≥ 1 and all j ≥ n ≥ 1, (4.10)

then Theorem 4.2.1 follows readily from Proposition 4.2.1. Clearly, if an = 0, n ≥ 1 and

bn|an|n1/q

is monotone increasing,

then it is quasi-monotone increasing. On the other hand, if an = 0, n ≥ 1 and

bn|an|n1/q

is monotone decreasing to 0,

then it is not quasi-monotone increasing. We will also provide below an example,

Example 4.2.3, wherein the conditions of Theorem 4.2.1 are satisfied but those of

Proposition 4.2.1 are not satisfied. To see that condition (4.10) indeed implies condition

74

(4.4), note that for all n ≥ 1, (4.10) yields

b1|a1|

≤ C infj≥1

bj|aj |j1/q

≤ C bn|an|n1/q

whence

|an|bn

≤ C |a1|b1

· 1n1/q, n ≥ 1. (4.11)

Thus condition (4.4) holds.

Proof of Theorem 4.2.1 with condition (4.4) replaced by condition (4.10).

It follows from (4.11) that

∞∑n=1

P∥anV ∥ > Dbn ≤∞∑n=1

P

∥V ∥ >

Db1C |a1|

n1/q

< ∞

since E∥V ∥q < ∞. Rewrite condition (4.10) as follows:

bj|aj |

≤ C bnj1/q

|an|n1/q, 1 ≤ j ≤ n, n ≥ 1. (4.12)

Then (max1≤j≤n

bj|aj |

) ∞∑j=n

|aj |bj

≤ C bn|an|

∞∑j=n

|aj |bj

(by (4.12))

= C

∞∑j=n

bn|an|

|aj |bj

≤ C∞∑j=n

Cn1/q

j1/q(by (4.10))

= C 2n1/q∞∑j=n

1

j1/q

= O(n)

as was shown in the proof of Theorem 4.2.1. The SLLN (4.3) follows from Proposition

4.2.1. 2

75

The following example, Example 4.2.3, shows that the conditions of Theorem 4.2.1

are satisfied but those of Proposition 4.2.1 are not satisfied.

Example 4.2.3. Let Vn, n ≥ 1 be a sequence of random elements in a real

separable Banach space X where Vn, n ≥ 1 is stochastically dominated by a random

element V in the sense that (2.7) holds for some constant D < ∞. Suppose that

E∥V ∥q < ∞ for some 0 < q < 1. Let an, n ≥ 1 and bn, n ≥ 1 be sequences of

constants satisfying 0 < bn ↑ ∞ and

an =

bn2n

for n odd,

bn4n

for n even.

Then (4.4) holds sinceanbn

≤ 12n= O

(1

n1/q

).

But (4.10) fails since, for n even,

bnann1/q

bn+1an+1(n + 1)1/q

=4n

n1/q(n + 1)1/q

2n+1

=

(n + 1

n

)1/q2n−1

→ ∞ as n even approaches ∞.

The SLLN (4.3) follows from Theorem 4.2.1. However, for n even,

max1≤j≤n+1

bj|aj |

n + 1

∞∑j=n+1

|aj |bj

≥ 4n

n + 1· 12n+1

=2n

2n + 2

→ ∞ as n even approaches ∞.

Thus condition (4.1) of Proposition 4.2.1 is not satisfied.

76

The following example, Examples 4.2.4, shows that the conditions of Proposition

4.2.1 are satisfied but those of Theorem 4.2.1 are not satisfied.

Example 4.2.4. Let α > 2 and let Vn, n ≥ 1 be a sequence of identically

distributed random variables with V1 having probability density function

f (x) =c

|x |α

α−1 (log |x |)3I[e,∞)(|x |), −∞ < x < ∞

where c is a positive constant. Let

an = log(n + 1), bn = nα−1, n ≥ 1.

Now we prove that conditions (4.1) and (4.2) hold as follows. Let Z(x) = log(x + 1) and

let u = −(α − 1). Then Z is varying slowly; that is, Z varies regularly with exponent

ρ = 0. Define Z ∗u (x) as in (2.5). By Lemma 2.3.2, Z ∗

u (x) < ∞ since u < −1. Then, by

Lemma 2.3.3 (i),

limx→∞

xu+1Z(x)

Z ∗u (x)

= −(u + ρ+ 1) = −u − 1 = α− 2 > 0.

Thus, (max1≤j≤n

bj|aj |

) ∞∑j=n

|aj |bj=

n(α−1)

log(n + 1)

∞∑j=n

log(j + 1)

jα−1

≤ nα−1

log(n + 1)

∫ ∞

n−1

log(y + 1)

yα−1dy

=Z ∗u (n − 1)nuZ(n)

= (1 + o(1))(n − 1)u+1Z(n − 1)(α− 2)nuZ(n)

= O(n).

Hence, condition (4.1) holds.

Furthermore, we prove that condition (4.2) holds with V = V1 and D = 1 as follows.

Let Z(x) = (log t)−3 and u = −α/(α − 1). Then Z is varying slowly; that is, Z varies

77

regularly with exponent ρ = 0. Define Z ∗u (x) as in (2.5). By Lemma 2.3.2, Z ∗

u (x) < ∞

since u < −1. Then, by Lemma 2.3.3 (i),

limx→∞

xu+1Z(x)

Z ∗u (x)

= −(u + ρ+ 1) = −u − 1 = 1

α− 1> 0.

Thus, we have

P|V1| > x = 2∫ ∞

x

f (y)dy = 2c

∫ ∞

x

y uZ(y)dy

= 2cZ ∗u (x) = (1 + o(1))2c(α− 1)xu+1Z(x)

which implies

P|V1| > x = (1 + o(1))2c(α− 1)x

1α−1 (log x)3

as x → ∞. (4.13)

Then, by (4.13),

P|anV1| > bn = P|V1| >

nα−1

log(n + 1)

= (1 + o(1))

2c(α− 1)(log(n + 1))1

α−1

n[(α− 1) log n − log log(n + 1)]3

= (1 + o(1))2c(α− 1)(log n)

1α−1

n(α− 1)3(log n)3

= (1 + o(1))2c

(α− 1)2n(log n)3α−4α−1.

Thus, condition (4.2) holds with V = V1 and D = 1 since α > 2 (hence (3α−4)/(α−1) >

1). Then, by Proposition 4.2.1, the SLLN∑ni=1(log(i + 1))Xi

nα−1→ 0 a. c.

holds.

To show that the hypotheses of Theorem 4.2.1 are not satisfied, note that condition

(4.4) holds for some q ∈ (0, 1) if and only if 1/q < α − 1; that is q > (α − 1)−1. On the

78

other hand, by (4.13),

∞∑n=1

P|V1| > n1/q =∞∑n=1

(1 + o(1))2c(α− 1)q3

n1

q(α−1) (log n)3< ∞

if and only if (q(α − 1))−1 ≥ 1; that is, q ≤ (α − 1)−1. Therefore, in Theorem 4.2.1,

condition (4.4) and the condition that E |V |q < ∞ for some 0 < q < 1 cannot both hold.

Thus Theorem 4.2.1 is not applicable for this example.

Remark 4.2.2. In Example 4.2.4, recall that 0 < (α − 1)−1 < q < 1 is necessary in

order for condition (4.4) to hold. In such a case,

bn|an|n1/q

is quasi-monotone increasing

sincebnann1/q

=nα−1−1/q

log(n + 1)

is monotone increasing. Therefore, condition (4.1) of Proposition 4.2.1 is satisfied by

the given “Proof of Theorem 4.2.1 with condition (4.4) replaced by condition (4.10)”. But

as was shown above, E |V |q = ∞ for q ∈ ((α − 1)−1, 1) and so Theorem 4.2.1 is not

applicable for Example 4.2.4.

The following example, Example 4.2.5, shows that the conditions of both Theorem

4.2.1 and Proposition 4.2.1 are satisfied. Consequently, Examples 4.2.3, 4.2.4, and

4.2.5 demonstrate that the conditions of Theorem 4.2.1 and Proposition 4.2.1 are

compatible with each other, but that neither set of conditions implies the other.

Example 4.2.5. Let Vn, n ≥ 1 be a sequence of random elements in a real

separable Banach space X . Suppose Vn, n ≥ 1 is stochastically dominated by a

random element V in the sense that (2.7) holds for some constant D < ∞. Let 0 < q < 1

and let

an = 1, bn = n1/q, n ≥ 1.

79

Suppose that E∥V ∥q < ∞. Let cn = n1/q for n ≥ 1. Then cpn /n = n1/q−1 ↑ and

lim infn→∞

cp2ncpn= lim inf

n→∞

(2n)1/q

n1/q= 21/q > 2

since 1/q > 1. Therefore, condition (4.1) holds since(max1≤j≤n

bj|aj |

) ∞∑j=n

|aj |bj=

(max1≤j≤n

cpj

) ∞∑j=n

1

cpj= O(n)

by Remark 3.2.3 taking r = 2. Furthermore (4.2) holds; i.e.,

∞∑n=1

P∥V ∥ > Dn1/q < ∞ (4.14)

since E∥V ∥q < ∞. In summary, all of the conditions of Proposition 4.2.1 hold.

Clearly, condition (4.4) holds. Hence, all of the conditions of Theorem 4.2.1 also hold.

Consequently, Theorem 4.2.1 and Proposition 4.2.1 can each be employed to establish

the SLLN (4.3).

The following theorem, Theorem 4.2.2, is a direct corollary of Theorem 4.2.1 in the

case where 0 < q < 1. There is a trade-off between the moment condition E∥V ∥q < ∞

and condition (4.15); the closer q is to 1, the moment condition E∥V ∥q < ∞ becomes

stronger whereas (4.15) becomes weaker. Theorem 4.2.2 is also a new result even

when the underlying Banach space is the real line R.

Theorem 4.2.2. Let Vn, n ≥ 1 be a sequence of random elements in a real

separable Banach space X . Suppose that Vn, n ≥ 1 is stochastically dominated by a

random element V in the sense that (2.7) holds for some constant D < ∞. Moreover,

suppose that E∥V ∥q < ∞ for some 0 < q ≤ 1. Let an, n ≥ 1 and bn, n ≥ 1 be

sequences of constants such that 0 < bn ↑ ∞,

∞∑n=1

(|an|bn

)q< ∞, (4.15)

and

0 <|an|bn

↓ if 0 < q < 1.

80

Then the SLLN (4.3) holds irrespective of the joint distributions of the Vn, n ≥ 1.

Proof. We first treat the case where 0 < q < 1. Note that if∑∞n=1 en is a convergent

series of positive constants with en ↓, then not only is en = o(1) but, also, nen = o(1)

(e.g., Knopp (1951, p. 124)). Hence, condition (4.15) implies condition (4.4). Thus, the

SLLN (4.3) follows immediately from Theorem 4.2.1.

Next, we treat the case where q = 1. Note that

E

(∞∑n=1

∥∥∥∥anVnbn∥∥∥∥)

=

∞∑n=1

E

∥∥∥∥anVnbn∥∥∥∥ (by Lemma 2.3.6)

=

∞∑n=1

|an|bnE ∥Vn∥

≤∞∑n=1

|an|bnD2E ∥V ∥ (by Lemma 2.3.10 taking t = 0)

<∞ (by E∥V ∥ < ∞ and (4.15)).

Thus∞∑n=1

∥∥∥∥anVnbn∥∥∥∥ < ∞ a. c.

whence, by Lemma 2.3.5 in the random variable case (the real line version of the

Kronecker lemma),∥∑ni=1 aiVi∥bn

≤∑ni=1 ∥aiVi∥bn

→ 0 a. c.

and so ∑ni=1 aiVibn

→ 0 a. c.

Hence we obtain the SLLN (4.3). 2

Remark 4.2.3. Taking an ≡ 1 and bn = n1/q, n ≥ 1 where 0 < q < 1, it is easily seen

that condition (4.4) of Theorem 4.2.1 holds but condition (4.15) of Theorem 4.2.2 fails.

The following example, Example 4.2.6, shows that Theorem 4.2.2 and Proposition

4.2.1 are compatible with each other, but that neither result implies the other.

81

Example 4.2.6. Let Vn, n ≥ 1 be a sequence of identically distributed random

elements in a real separable Banach space X . Suppose E∥V1∥q < ∞ for some

0 < q ≤ 1. Let an ≡ 1, n ≥ 1. Take bn = nα, n ≥ 1. Then condition (4.15) holds

if and only if α > 1/q, and condition (4.1) holds if and only if α > 1. Therefore, if

α > 1/q, then both conditions (4.15) and (4.1) hold. Furthermore, E∥V1∥q < ∞

implies E∥V1∥1/α < ∞ since q > 1/α. Hence, condition (4.2) holds with V = V1 and

D = 1. In this case, the conditions of both Theorems 4.2.2 and Proposition 4.2.1 are

satisfied. If we instead let bn = n1/q, n ≥ 1 where now 0 < q < 1, then condition (4.1)

holds since 1/q > 1. Moreover, condition (4.2) holds with V = V1 and D = 1 since

E∥V1∥q < ∞. However, condition (4.15) fails. Thus when bn = n1/q, n ≥ 1, the conditions

of Proposition 4.2.1 are satisfied but those of Theorem 4.2.2 are not. Next let q = 1 and

bn = n(log(n + 1))α, n ≥ 1 where α > 1. Then condition (4.15) holds. However,

∞∑j=n

1

bj= O

(1

(log(n + 1))α−1

)= O

(n

bn

)and so condition (4.1) fails. Thus when bn = n(log(n + 1))α, n ≥ 1 where α > 1, the

conditions of Theorem 4.2.2 are satisfied but those of Proposition 4.2.1 are not.

The following corollary, Corollary 4.2.2, follows immediately from Theorem 4.2.1

(when 0 < q < 1) and from Theorem 4.2.2 (when q = 1).

Corollary 4.2.2. Let Vn, n ≥ 1 be a sequence of random elements in a real

separable Banach space X . Suppose that Vn, n ≥ 1 is stochastically dominated

by a random element V in the sense that (2.7) holds for some constant D < ∞. Let

an, n ≥ 1 and bn, n ≥ 1 be sequences of constants such that an = 0, n ≥ 1 and

0 < bn ↑ ∞. Suppose for some 0 < q ≤ 1 that E∥V ∥q < ∞,

anbn= O

(1

n1/q

)if 0 < q < 1,

and∞∑n=1

|an|bn

< ∞ if q = 1.

82

Then the SLLN (4.3) holds irrespective of the joint distributions of the Vn, n ≥ 1.

The following theorems, Theorems 4.2.3 and 4.2.4, provide sets of conditions under

which the SLLN (4.3) holds where the random elements Vn, n ≥ 1 are stochastically

dominated but are not necessarily independent. Theorems 4.2.3 and 4.2.4 are new

results even when the underlying Banach space is the real line R.

Theorem 4.2.3. Let Vn, n ≥ 1 be a sequence of random elements in a real

separable Banach space X . Suppose that Vn, n ≥ 1 is stochastically dominated

by a random element V in the sense that (2.7) holds for some constant D < ∞. Let

an, n ≥ 1 and bn, n ≥ 1 be sequences of constants such that an = 0, n ≥ 1,

0 < bn ↑ ∞, bn/|an| ↑, and

bn|an|nL(n)

is quasi-monotone increasing ;

that is,

bn|an|nL(n)

≤ C bj|aj |jL(j)

< ∞ for some C ≥ 1 and all j ≥ n ≥ 1 (4.16)

where L : [1,∞) → (0,∞) is a nondecreasing slowly varying function. Set a0 = a1 and

b0 = 0. Suppose that

∞∑n=1

nL(n)

(∞∑j=n

1

jL(j)

)P

bn−1|an−1|

< ∥V ∥ ≤ bn|an|

< ∞. (4.17)

Then the SLLN (4.3) holds irrespective of the joint distributions of the Vn, n ≥ 1.

Remark 4.2.4. (i) It follows from (4.16) that

|an|bn

≤ C |a1|b1

· L(1)nL(n)

, n ≥ 1

and so

anbn= O

(1

nL(n)

)= O

(1

n

)(4.18)

83

implying

bn|an|

↑ ∞ (4.19)

since bn/|an| ↑.

(ii) Condition (4.17) implies that

∞∑n=1

P∥anV ∥ > bn < ∞. (4.20)

To see this, note that (4.17) ensures that

∞∑j=n

1

jL(j)< ∞ (4.21)

for each n ≥ 1 unless V = 0 a. c. (in which case (4.2) and (4.3) are immediate). LetZ(x) = 1/L(x) and let u = −1. Then Z is varying slowly; that is, Z varies regularlywith exponent ρ = 0. Define Z ∗

u (x) as in (2.5). Then Z ∗u (x) < ∞ by (4.21). Hence,

by Lemma 2.3.3 (i),

limx→∞

xu+1Z(x)

Z ∗u (x)

= −(u + ρ+ 1) = 0.

Thus,

limn→∞L(n)

∞∑j=n

1

jL(j)=∞

whence (4.17) implies that

∞∑n=1

nP

bn−1|an−1|

< ∥V ∥ ≤ bn|an|

< ∞.

This is readily seen via (4.19) to be equivalent to (4.20).

(iii) It follows from (4.21) that L(n) ↑ ∞ and so (4.18) can be strengthened to

anbn= O

(1

nL(n)

)= o

(1

n

).

Thus (4.21) ensures thatbnn|an|

→ ∞.

Proof of Theorem 4.2.3. As was noted in Remark 4.2.4 (ii), condition (4.17) implies

(4.20) and recalling that Vn, n ≥ 1 is stochastically dominated by V in the sense that

84

(2.7) holds for some constant D < ∞, we have

∞∑n=1

P∥anVn∥ > Dbn ≤∞∑n=1

DP ∥anDV ∥ > Dbn < ∞,

which, by Lemma 2.3.4, ensures that

Plim infn→∞

[I (∥anVn∥ > Dbn) = 0]= 1,

Therefore,∞∑n=1

∥anVn∥I (∥anVn∥ > Dbn) < ∞ a. c.

Since 0 < bn ↑ ∞,1

bn

n∑i=1

∥aiVi∥I (∥aiVi∥ > Dbi)→ 0 a. c.

Hence, to prove the SLLN (4.3), it suffices to show that

1

bn

n∑i=1

∥aiVi∥I (∥aiVi∥ ≤ Dbi)→ 0 a. c.

which, by Lemma 2.3.5 in the random variable case (the real line version of the

Kronecker lemma), will hold once we show that

∞∑n=1

1

bn∥anVn∥I (∥anVn∥ ≤ Dbn) < ∞ a. c.

Hence, it suffices to verify that

∞∑n=1

1

bnE [∥anVn∥I (∥anVn∥ ≤ Dbn)] < ∞. (4.22)

Note, by Lemma 2.3.11 taking q = 1, that

∞∑n=1

1

bnE [∥anVn∥I (∥anVn∥ ≤ Dbn)]

≤D2∞∑n=1

P ∥anV ∥ > bn+D2∞∑n=1

|an|bnE [∥V ∥I (∥anV ∥ ≤ bn)] .

85

Then recalling (4.20), (4.22) will follow provided we can show that

∞∑n=1

|an|bnE [∥V ∥I (∥anV ∥ ≤ bn)] < ∞. (4.23)

First, it follows from (4.16) that, for all n ≥ j ≥ 1,

bj|aj |

|an|bn

≤ C jL(j)nL(n)

. (4.24)

Then, to verify (4.23), note that

∞∑n=1

|an|bnE [∥V ∥I (∥anV ∥ ≤ bn)]

=

∞∑n=1

|an|bn

n∑j=1

E

[∥V ∥I

(bj−1|aj−1|

< ∥V ∥ ≤ bj|aj |

)](since bn/|an| ↑)

≤∞∑n=1

|an|bn

n∑j=1

bj|aj |P

bj−1|aj−1|

< ∥V ∥ ≤ bj|aj |

=

∞∑j=1

P

bj−1|aj−1|

< ∥V ∥ ≤ bj|aj |

∞∑n=j

bj|aj |

|an|bn

≤C∞∑j=1

P

bj−1|aj−1|

< ∥V ∥ ≤ bj|aj |

∞∑n=j

jL(j)

nL(n)(by (4.24))

=C

∞∑n=1

nL(n)

(∞∑j=n

1

jL(j)

)P

bn−1|an−1|

< ∥V ∥ ≤ bn|an|

<∞ (by (4.17))

proving (4.23) and yielding the conclusion (4.3). 2

In the following theorem, Theorem 4.2.4, there is a trade-off between the moment

condition E∥V1∥q < ∞ and condition (4.25); the closer q is to 1, the moment condition

E∥V1∥q < ∞ becomes stronger whereas (4.25) becomes weaker.

Theorem 4.2.4. Let Vn, n ≥ 1 be a sequence of random elements in a real

separable Banach space X . Suppose that Vn, n ≥ 1 is stochastically dominated

by a random element V in the sense that (2.7) holds for some constant D < ∞. Let

an, n ≥ 1 and bn, n ≥ 1 be sequences of constants such that an = 0, n ≥ 1,

86

0 < bn ↑ ∞ and bn/|an| ↑. Suppose for some 0 ≤ q ≤ 1 that E∥V ∥q < ∞ and

∞∑j=n

|aj |bj= O

((|an|bn

)1−q). (4.25)

If (4.20) holds, then the SLLN (4.3) holds irrespective of the joint distributions of the

Vn, n ≥ 1.

Proof. Note at the outset that bn/|an| ↑ ∞. Proceeding as in the proof of Theorem 4.2.3,

it suffices to show that (4.23) holds. To this end, set a0 = 1 and b0 = 0. Then

∞∑n=1

|an|bnE [∥V ∥I (∥anV ∥ ≤ bn)]

=

∞∑n=1

|an|bn

n∑j=1

E

[∥V ∥I

(bj−1|aj−1|

< ∥V ∥ ≤ bj|aj |

)](since bn/|an| ↑)

=

∞∑j=1

(∞∑n=j

|an|bn

)E

[∥V ∥I

(bj−1|aj−1|

< ∥V ∥ ≤ bj|aj |

)]

≤C∞∑j=1

(|aj |bj

)1−qE

[∥V ∥I

(bj−1|aj−1|

< ∥V ∥ ≤ bj|aj |

)](by (4.25))

=C

∞∑j=1

(|aj |bj

)1−qE

[∥V ∥q∥V ∥1−qI

(bj−1|aj−1|

< ∥V ∥ ≤ bj|aj |

)]

≤C∞∑j=1

E

[∥V ∥qI

(bj−1|aj−1|

< ∥V ∥ ≤ bj|aj |

)]=CE∥V ∥q (by Lemma 2.3.6 and bn/|an| ↑ ∞)

<∞

proving (4.22) and yielding the conclusion (4.3). 2

Remark 4.2.5. (i) When q = 1, (4.25) of Theorem 4.2.4 is equivalent to(4.15). Thus, when q = 1, Theorem 4.2.4 follows readily from Theorem 4.2.2, and(4.20) can be dispensed with as an assumption. In fact, the condition E∥V ∥ < ∞implies (4.20) since, by the Markov inequality and recalling (4.25)

∞∑n=1

P∥anV ∥ > bn ≤ E∥V ∥∞∑n=1

(|an|bn

)< ∞.

87

(ii) When q = 0, condition (4.25) of Theorem 4.2.4 implies (4.1) since

bn|an|

∞∑j=n

|aj |bj= O(1) = O(n).

In this case, the condition E∥V ∥q < ∞ is automatic, and (4.20) cannot bedispensed with.

(iii) Example 3.1 of Rosalsky and Stoica (2010) satisfies the hypotheses ofTheorem 4.2.3 but not the hypotheses of Theorem 4.2.1, Proposition 4.2.1,Theorem 4.2.2, or Theorem 4.2.4.

(iv) Example 3.2 of Rosalsky and Stoica (2010) satisfies the hypotheses ofTheorem 4.2.1, Proposition 4.2.1, and Theorem 4.2.4 but not the hypotheses ofTheorem 4.2.2. If, in this example, X1 has probability density function

f (x) =c

xp+1(log x)αI[e,∞)(x), −∞ < x < ∞

where 0 < p < 1, α > 2 and c is a constant, then taking

L(x) = (log x)α−1, x ≥ 1,

the hypotheses of Theorem 4.2.3 are also satisfied.

Now we consider sets of conditions under which the particular SLLN of the form∑ni=1 Vi

bn→ 0 a. c. (4.26)

holds where bn, n ≥ 1 is a sequence of positive constants satisfying 0 < bn ↑ ∞

and the random elements Vn, n ≥ 1 are identically distributed but are not necessarily

independent. We note that taking bn = n1/q, n ≥ 1 where 0 < q < 1, Theorem 4.2.1

demonstrates that the independence hypothesis in the Marcinkiewicz-Zygmund type

SLLN can be dispensed with when 0 < q < 1. Proposition 4.2.1 likewise demonstrates

this upon noting that∞∑j=n

1

j1/q= O(n1−1/q)

when 0 < q < 1. Earlier proofs that the real line version of the Marcinkiewicz-Zygmund

SLLN holds without the independence hypothesis when 0 < q < 1 may be found in

Sawyer (1966), Chatterji (1970), and Martikainen and Petrov (1980).

88

The following corollaries, Corollaries 4.2.3, 4.2.4, and 4.2.5, are obtained from

Theorems 4.2.2, 4.2.3, and 4.2.4, respectively. When the underlying Banach space X

is the real line R, Corollaries 4.2.3, 4.2.4, and 4.2.5 were established by Petrov (1973,

Theorem 1), Rosalsky and Stoica (2010, Theorem 2.1), and Rosalsky and Stoica (2010,

Theorem 2.2), respectively.

Corollary 4.2.3. Let Vn, n ≥ 1 be a sequence of identically distributed random

elements in a real separable Banach space X . Suppose E∥V1∥q < ∞ for some

0 < q ≤ 1. If bn, n ≥ 1 is a nondecreasing sequence of positive constants such that

∞∑n=1

1

bqn< ∞, (4.27)

then the SLLN (4.26) holds irrespective of the joint distributions of the Vn, n ≥ 1.

Proof. Take an ≡ 1, n ≥ 1 in Theorem 4.2.2. Then the conclusion (4.26) follows. 2

Corollary 4.2.4. Let Vn, n ≥ 1 be a sequence of identically distributed random

elements in a real separable Banach space X . Let b0 = 0 and bn, n ≥ 1 be a

nondecreasing sequence of positive constants such that

bnnL(n)

→ ∞ (4.28)

where L : [1,∞)→ (0,∞) is a nondecreasing slowly varying function. Suppose that

bnnL(n)

is quasi-monotone increasing ;

that is,

bnnL(n)

≤ C bjjL(j)

< ∞ for some C ≥ 1 and all j ≥ n ≥ 1

If

∞∑n=1

nL(n)

(∞∑j=n

1

jL(j)

)Pbn−1 < ∥V1∥ ≤ bn < ∞,

then the SLLN (4.3) holds irrespective of the joint distributions of the Vn, n ≥ 1.

89

Proof. Note that (4.28) ensures that 0 < bn ↑ ∞. Take an ≡ 1, n ≥ 1 in Theorem 4.2.3.

Then the conclusion (4.26) follows. 2

Corollary 4.2.5. Let Vn, n ≥ 1 be a sequence of identically distributed random

elements in a real separable Banach space X . Let bn, n ≥ 1 be a nondecreasing

sequence of constants such that, for some 0 ≤ q ≤ 1,

∞∑j=n

1

bj= O

(1

b1−qn

).

Suppose that E∥V1∥q < ∞ and

∞∑n=1

P∥V1∥ > bn < ∞.

Then the SLLN (4.26) holds irrespective of the joint distributions of the Vn, n ≥ 1.

Proof. Take an ≡ 1, n ≥ 1 in Theorem 4.2.4. Then the conclusion (4.26) follows. 2

90

CHAPTER 5FUTURE RESEARCH AND CONCLUSIONS

5.1 Future Research

Some thoughts concerning future research will now be discussed.

The first idea about future research is that whether Theorem 4.2.4 can be improved.

Note that in Theorem 4.2.4, there are in effect two moment conditions, namely condition

(4.20) and the condition E∥V ∥q < ∞, both of which we applied to prove Theorem

4.2.4. What is the relationship between them? Do they imply each other or one strictly

stronger than the other? Or are they not comparable in general? We now present two

examples, Examples 5.1.1 and 5.1.2, that satisfy all the hypotheses of Theorem 4.2.4.

In these two examples, the hypothesis E∥V ∥q < ∞ is strictly stronger than (4.20). We

need additional examples to clarify the relationship between the condition (4.20) and the

condition E∥V ∥q < ∞ apropos of Theorem 4.2.4.

Example 5.1.1. Let Vn, n ≥ 1 be a sequence of identically distributed random

variables. Let 0 < α < 1 and

an = 1, bn = n1/α, n ≥ 1.

Then (bn|an|

)1−q ∞∑j=n

|aj |bj= n

1−qα

∞∑j=n

1

n1/α= n

1−qα O(n1−

1α ) = O(n1−

qα ).

Thus, (4.25) holds if and only if q ≥ α. Moreover, (4.20) holds if and only if E |V1|α < ∞.

Therefore, all of the hypotheses of Theorem 4.2.4 are satisfied if q ≥ α. Furthermore,

E |V1|q < ∞ implies E |V1|α < ∞ for q ≥ α; i.e., E |V1|q < ∞ implies (4.20) for q ≥ α.

However, (4.20) does not necessarily imply E |V1|q < ∞ if q > α.

Example 5.1.2. Let Vn, n ≥ 1 be a sequence of identically distributed random

variables with V1 having probability density function

f (x) =c

|x |1+β(log |x |)3I[e,∞)(|x |), −∞ < x < ∞

91

where 0 < β < 1 and c is a positive constant. Let 0 < α < 1 and

an = log(n + 1), bn = n1/α, n ≥ 1.

Then by Lemma 2.3.3 (i),

P|V1| > x = 2∫ ∞

x

f (y)dy = 2c

∫ ∞

x

y−(1+β)(log y)−3dy

= (1 + o(1))2c

βxβ(log x)3as x → ∞

and∞∑j=n

|aj |bj=

∞∑j=n

log(j + 1)

j1/α= (1 + o(1))

α

1− α

log(n + 1)

n1−αα

.

Thus, (bn|an|

)1−q ∞∑j=n

|aj |bj= (1 + o(1))C

(n1/α

log(n + 1)

)1−qlog(n + 1)

n1−αα

= (1 + o(1))C[log(n + 1)]q

nq−αα

,

(5.1)

P|V1| > n1/q = (1 + o(1))C

xβ/q(log n)3, (5.2)

and

P|anV1| > bn = P|V1| >

n1/α

log(n + 1)

= (1 + o(1))

C

nβ/α(log n)3−β. (5.3)

By (5.1), (4.25) holds if and only if α < q. By (5.2), E |V1|q < ∞ if and only if q ≤ β.

By (5.3), (4.20) holds if and only if α ≤ β. Therefore, all of the hypotheses of Theorem

4.2.4 are satisfied if α < q ≤ β. Furthermore, E |V1|q < ∞ and α ≤ q implies α ≤ β;

i.e., E |V1|q < ∞ and α ≤ q implies (4.20). However, (4.20) does not necessarily imply

E |V1|q < ∞ since α ≤ β < q can hold.

Since convergence almost certainly implies convergence in probability, it is natural

to raise the second thought about future research as to whether the weak law of large

numbers (WLLN) can be obtained under hypotheses which are strictly weaker than

92

those in our main results. We would hope that the conditions of the main theorems in

Chapter 3 and 4 can be strictly weakened so that the corresponding SLLN fails but the

WLLN holds.

The third thought concerning future research is motivated by complete convergence.

A sequence of random variables Xn, n ≥ 1 is said to converge completely to 0 if

∞∑n=1

P|Xn| ≥ ε < ∞ for all ε > 0.

This kind of convergence was introduced by Hsu and Robbins (1947). It is easily

seen by Lemma 2.3.4 (the Borel-Cantelli lemma) that complete convergence to 0

implies almost certain convergence to 0, and the converse is true if the Xn, n ≥ 1 is

independent. We would hope that the assumptions of the main theorems in Chapter 3

and 4 can be strengthened to achieve complete convergence results.

Finally, we would consider to obtain SLLNs for double sums of random elements

Vi ,j , i ≥ 1, j ≥ 1 of the form∑ni=1

∑mj=1 Vi ,j

bm,n→ 0 a. c. as m ∧ n → ∞.

or ∑ni=1

∑mj=1 Vi ,j

bm,n→ 0 a. c. as m ∨ n → ∞

respectively, where bm,n, m ≥ 1, n ≥ 1 is an array of positive constants with

bm,n → ∞ as m ∧ n → ∞

or

bm,n → ∞ as m ∨ n → ∞,

respectively. We also would like to consider obtaining complete convergence results for

double sums of random elements Vi ,j , i ≥ 1, j ≥ 1.

93

5.2 Conclusions

In this dissertation we have presented results pertaining to the SLLN problem for

sums of Banach space valued random elements and the main results are new even

when the underlying Banach space is the real line.

In Chapter 3, we establish Theorem 3.2.1, a very general SLLN for normed

weighted sums of independent Banach space valued random elements which are

not necessarily identically distributed. The underlying Banach space is assumed to be of

Rademacher type p (1 ≤ p ≤ 2) and the sequence of random elements is assumed to

be stochastically dominated by a random element. Special cases of Theorem 3.2.1 are:

(i) Proposition 3.2.1 which is Theorem 1 of Adler, Rosalsky, and Taylor (1989)

(ii) Theorem 3.2.2 which is an improved version of Theorem 6 of Adler, Rosalsky, andTaylor (1989).

Theorem 3.2.2 contains Proposition 3.2.2 which is the result of Sung (1997). (Theorem

6 of Adler, Rosalsky, and Taylor (1989) does not contain Sung’s (1997) result.) Theorem

3.2.2 also contains Theorem 4.1 of Woyczynski (1980) (Remark 3.2.6 (iv)) and Theorem

5 of Teicher (1985) (Corollary 3.2.1). Theorems 3.2.1 and 3.2.2 are new even when the

underlying Banach space is the real line.

In Chapter 4, we obtain SLLNs irrespective the joint distributions of the random

elements and no geometric conditions are imposed on the underlying Banach space.

We established four theorems (Theorems 4.2.1, 4.2.2, 4.2.3, and 4.2.4) all of which have

the assumption that the sequence of random elements is stochastically dominated by a

random element. Theorem 4.2.1 is a “0 < q < 1” version of Theorem 3.2.2. Theorem

4.2.2 pertains to 0 < q ≤ 1 and the 0 < q < 1 part follows from Theorem 4.2.1.

Theorems 4.2.3 and 4.2.4 are obtained separately. Theorems 4.2.1, 4.2.2, 4.2.3, and

4.2.4 are new even when the underlying Banach space is the real line. Special cases of

these four theorems are:

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(i) Corollary 4.2.1 which is a direct corollary of Theorem 4.2.1 and is a new versionof Theorem 5 of Teicher (1985) and Corollary 2 of Sung (1997) which pertained tosequences of i.i.d. Lp random variables where 1 ≤ p < 2

(ii) Corollary 4.2.2 which follows immediately from Theorem 4.2.1 (when 0 < q < 1)and from Theorem 4.2.2 (when q = 1)

(iii) Corollary 4.2.3 which follows immediately from Theorem 4.2.2 and is Theorem 1 ofPetrov (1973) when the underlying Banach space is the real line

(iv) Corollary 4.2.4 which follows immediately from Theorem 4.2.3 and is Theorem 2.1of Rosalsky and Stoica (2010) when the underlying Banach space is the real line

(v) Corollary 4.2.5 which follows immediately from Theorem 4.2.4 and is Theorem 2.2of Rosalsky and Stoica (2010) when the underlying Banach space is the real line.

In Chapters 3 and 4, we have presented various examples regarding distinct

aspects of the results obtained in the current work. Examples are provided which

illustrate the results and which demonstrate their sharpness and distinctions. For some

of the aforementioned corollaries, we presented examples illustrating that these results

are indeed extended by the main theorems of the current work.

We also discussed possible ideas concerning future research in the above section.

Over the past twenty years, a large literature of investigation has emerged on the WLLN,

on the complete convergence problem, and on limit theorems for double sums. Many

of the results concern Banach space valued summands. The starting point for a new

investigation should be a literature search using Mathematical Reviews on the Web

(MathSciNet) [http://www.ams.org/mathscinet/].

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BIOGRAPHICAL SKETCH

Yuan Liao was born in 1980 in Changchun, China. Upon graduation from the high

school affiliated with the Northeast University in China in July 1999, he enrolled as an

undergraduate student in the Department of Mathematics at the University of Science

and Technology of China where he earned a Bachelor of Arts Degree in Mathematics

in July 2004. In September 2004, he entered a masters program in statistics in the

Department of Mathematics at the Graduate University of the Chinese Academy of

Sciences where he earned a Master of Arts Degree in Statistics in July 2007. In August

2007, he entered a Ph.D. program in the Department of Statistics at the University

of Florida. During his graduate education at University of Florida, he was appointed

as a teaching assistant for different classes in the Department of Statistics. His main

research interests are Probability Theory and Limit Theory for Banach Space Valued

Random Elements.

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