almost sure and complete convergence of randomly weighted sums of independent random elements in...

7/25/2019 ALMOST SURE AND COMPLETE CONVERGENCE OF RANDOMLY WEIGHTED SUMS OF INDEPENDENT RANDOM ELEMENTS IN BANACH SPACES

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TAIWANESE JOURNAL OF MATHEMATICSVol. 15, No. 4, pp. 1759-1781, August 2011

This paper is available online at http://www.tjm.nsysu.edu.tw/

ALMOST SURE AND COMPLETE CONVERGENCE OF RANDOMLY

WEIGHTED SUMS OF INDEPENDENT RANDOM ELEMENTS IN

BANACH SPACES

Le Van Thanh1 and G. Yin2

Abstract. This work develops almost sure and complete convergence of

randomly weighted sums of independent random elements in a real separableBanach space. Sufficient conditions for almost sure convergence and complete

convergence in the sense defined by Hsu and Robbins are provided. Examples

showing that the conditions cannot be removed or weakened are given. It is

also demonstrated that some of the known theorems in the literature are special

cases of our results.

1. INTRODUCTION

According to Hsu and Robbins [9], a sequence of real-valued random variables

{Xn, n 1} converges completely to 0 if,

(1.1)

n=1

P

|Xn| >

< for all > 0.

Based on this definition, Hsu and Robbins [9] and Erdos [7] derived a necessary

and sufficient condition for a sequence of independent identically distributed (i.i.d.)

random variables{Xn, n 1}. The assertion is: EX1 = 0 andE|X1|2 < if

and only if

(1.2)

n=1

P n

i=1

Xi> n< for all >0.

Received April 12, 2010, accepted April 12, 2010.

Communicated by Jen-Chih Yao.2000 Mathematics Subject Classification: 60F15, 60B11, 60B12.

Key words and phrases: Almost sure convergence, Complete convergence, Random weight, Row-wise

independent, Banach space.1Research of this author was supported in part by the National Science Foundation under DMS-

0603287.2 Research of this author was supported in part by the National Science Foundation under DMS-

0907753, and in part by the Air Force Office of Scientific Research under FA9550-10-1-0210.

1759


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1760 Le Van Thanh and G. Yin

Their theorem was extended by Baum and Katz [4] to establish a rate of convergence

in the sense of Marcinkiewicz-Zygmunds strong law of large numbers, which reads:

Let > 1/2, p 1, and{Xn, n 1} be a sequence of i.i.d. random variables.ThenE X1= 0 andE|X1|

p < if and only if

(1.3)

n=1

np2P n

i=1

Xi> n< for all > 0.

The results have been generalized and extended in several directions. A host of

researchers considered the problem in a Banach space setting; see Ahmed et al. [3],

Hu et al. [11], Sung [21], Hernandez et al. [8], Sung et al. [19, 22], and Wang[25] among others.

Recall that a sequence of random elements {Vni, n 1, 1 i n} is stochas-tically dominatedby a random elementV if for some constant C < ,

(1.4) P{Vni > t} CP{V > t}, t 0, n 1, 1 i n.

Using stochastic dominance, Hu et al. [10] treated triangular arrays of row-wise

independent random variables and obtained the following complete convergence

result with a Marcinkiewicz-Zygmund type normalization. Let{Xni, n 1, 1 i n} be an array of row-wise independent mean 0 random variables that isstochastically dominated by a random variableX, that is, condition (1.4) is satisfied.

IfE|X|2t < where1 t n1/t< for all > 0.

Subsequently, Wang et al. [25] considered triangular arrays of row-wise independent

random elements and obtained the following convergence rate. Let1/2 < 1and{Vni, n 1, 1 i n} be an array of row-wise independent random elements,which is stochastically dominated by a random elementVin the sense of condition(1.4) being satisfied. IfEVp < for some p >1 and

(1.6) maxkn

Pk

i=1 Vni> n 0 as n for all >0,

then

(1.7)

n=1

np2P n

i=1

Vni> n< for all >0.

Later, Sung established a complete convergence for weighted sums of random ele-

ments in [20]. Let{Vni, n 1, 1 i n} be an array of row-wise independentrandom elements that is stochastically dominated by a random element V. Letp 1and{bni, n 1, 1 i n} be an array of constants satisfying


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Randomly Weighted Sums in Banach Spaces 1761

(1.8) max1in

|bni| =O(n1/p)

and

(1.9)

ni=1

b2ni= o 1

log n

.

IfEV2p < and

(1.10)

n

i=1bniVni 0 in probability asn ,

then

(1.11)

n=1

P n

i=1

bniVni> < for all >0.

A number of researchers considered the weighted sums of random elements

where the weights are constants; see Ahmed et al. [3], Hu et al. [11], and Sung

[21]. In [16], Li et al. improved the main result of Thrum [24]; their result

reads: Let {Vi, i 1} be a sequence of independent identically distributed randomelements with EV1

p < for some p 1. Let {ani, n 1, 1 i n} be an

array of constants such that supn,i |ani|< and

(1.12)

ni=1

a2ni= O(n) for some 0 < < min{1, 2/p}.

If

(1.13) 1

n1/p

ni=1

aniVi 0 in probability,

then

(1.14) 1

n1/p

n

i=1 aniVi 0 a.s.In this paper, we obtain the almost sure convergence and complete convergence

for randomly weighted sums of arrays of row-wise independent random elements.

Although Cuzick [5], Li and Tomkins [18] and Li et al. [17] also considered almost

sure convergence with random weights, to the best of our knowledge, there has yet

been any result on the complete convergence of the case in which the weights are

random variables. Moreover, the results in the aforementioned papers do not imply

our results even in the case the weights are non-random. Most of the results of the

aforementioned work in the literature can be covered and improved by our results.


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As a motivation, we describe a problem arising in systems theory. Recall that

a transfer function is commonly used in the analysis of single-input single-output

systems, in signal processing, communication theory, and control theory. Consider a

continuous-time, linear, time-invariant system. Then a transfer function is the ratio

of the Laplace transforms of the output and input. It gives input/output character-

istics in the frequency domain. In certain applications, the underlying systems need

to be identified, which can be carried out by estimating the transfer functions using

observable data. Suppose that we wish to identify an unknown transfer function

K(). The input is given by z(), sampling interval is , and output (at sampling

time n) isyn, where

yn =

10

K(u)z(n u)du + n,

where {n} is a sequence of observation noise, which is a sequence of i.i.d. randomvariables with 0 mean and finite variance and is independent of z(). To identifyor to estimate K(), we treat this as a stochastic optimization problem in L2[0, 1],which is the Hilbert space consisting of square integral functions defined on the

interval [0, 1]. Let

x, y

and ||x||= [

x, x

]1/2 denote the inner product and thenorm onL2[0, 1], respectively. To carry out the desired task, we construct algorithm

of the form

(1.15) Kn+1(t) =Kn(t) +nz(nt)yn 10Kn(u)z(n u)du.

The above algorithm is stochastic approximation type analogs to its finite dimen-

sional counterpart [14]. Define Kn(t) =Kn(t) K(t), and useKn, Kn, K,andzn denote the random variables whose point values areK(t), Kn(t), K(t),andzn(n t), resp.

(1.16) Kn+1(t) =Kn(t) + nzn[Kn, zn +n].In the implementation, one often wishes to use a projection algorithm

(1.17)

Kn+1(t) =

Kn(t) + nzn[

Kn, zn

+ n]

,

where () is the projection to the unit ball in L2[0, 1]. As explained in [13, p.782], this can be further rewritten as

(1.18) Kn+1(t) =Kn(t) +nzn[Kn, zn +n] ncnKn+On,wherecn satisfiesncn< 1, andOn satisfies

m(T)j=0 Oj 0, for any 0 < T <

with m(t) = max{n : tn t} and tn =n1

j=0j . In analyzing such algorithms,one often needs to deal with a term (see [13, p. 783])

nk=0

nj=k+1

(1 j cj)

kzk(

zk, Kk+ k).


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As can be seen that this term is of the form a weighted sum of random variables.

Our work in this paper provides a sufficient condition for almost sure convergence

and complete convergence of such a sequence.

We also note that many linear models in statistics based on a random sample in-

volve weighted sums of dependent random variables. Examples include least-squares

estimators, nonparametric regression function estimators, and jackknife estimates,

among others. In this respect, the study of convergence for these weighted sums

has impact on statistics, probability, and their applications.

The rest of the paper is arranged as follows. Section 2 presents some definitions

and lemmas to be used in the proof of our results. Section 3 provided conditions foralmost sure convergence for randomly weighted sums. Our results in this section

improve the results by Li et al. [16]. Section 4 establishes complete convergence

for randomly weighted sums of arrays of row-wise independent random elements in

a real separable Banach space. In the main results, Theorem 3.1, Theorem 3.3, and

Theorem 4.1, no assumptions are made concerning the geometry of the underlying

Banach space. Finally, in Section 5, some corollaries and examples are presented

to illustrate the results. It demonstrates that our conditions are sharp and our results

lead to better convergence rates. Throughout this paper, the symbol C denotes a

generic positive constant (0< C < ) whose values may be different for differentappearances.

2. PRELIMINARIES

We begin with definitions, notation, and preliminary results prior to establishing

the main results. Let X be a real separable Banach space with norm ||||. A randomelement inXwill be denoted by V or by Vn, Wn, etc.

The expected value or mean of a random element V, denotedE V, is definedto be the Pettis integralprovided it exists. That is, Vhas expected value E V Xifh(EV) = E(h(V)) for every h X, whereX denotes the (dual) space of allcontinuous linear functionals on X. IfE||V||< , then (see, e.g., Taylor [23, p.40])Vhas an expected value. But the expected value can exist whenE||V|| = .

For an example, see Taylor [23, p. 41].Lemma 2.1 below is a Marcinkiewicz type inequality for some of independent

random elements; see de Acosta [1] for a proof.

Lemma 2.1. Let{Vni, n 1, 1 i n} be an array of row-wise independentrandom elements. Then for every r 1, there is a positive constantCr dependingonly on r such that for alln 1,

(2.19) E n

i=1

Vni En

i=1

Vnir Cr n

i=1

EVnir, for1 r 2,


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and

(2.20)

E n

i=1

Vni En

i=1

Vnir

Cr

ni=1

EVni2r/2

+n

i=1

EVnir

, forr >2.

The following lemma is a Banach space version of Lemma 2.1 of Li at al. [16].

The proof is similar to that of Lemma 2.2 in Hu at al. [11].

Lemma 2.2. Letr 1, and{Vn, n 1} be a sequence of random elementsand{V

n, n 1} an independent copy of{Vn, n 1}. IfVn 0 in probability,then

(i) Vn 0 a.s. if and only ifVn V

n 0 a.s.

(ii)

n=1nr2P{Vn }< for every >0 if and only if

n=1

nr2P{Vn

V

n } < for every > 0.

The third lemma due to Hu at al. [11] concerning the relationship between

convergence in probability and mean convergence for sums of independent bounded

random elements.

Lemma 2.3. Let{Vni, n 1, 1 i n} be an array of row-wise independentsymmetric random elements and suppose there exists > 0 such thatVni a.s. for alln 1, 1 i n. If

ni=1 Vni 0 in probability as n , then

En

i=1 Vni 0 asn .

The following lemma is a result of Adler at al. [2] concerning the expectation

of the truncation random elements that are stochastically dominated by a random

elementV.

Lemma 2.4. Let{Vni, n 1, 1 i n} be an array of random elements thatis stochastically dominated by a random elementV . Then for all p >0

(2.21)

E

VnipI(Vni t)

CtpP(V > t)+CE

VpI(V t)

, t 0, n 1, 1 i n,

and

(2.22)

E

VniI(Vni> t)

CE

VI(V >t)

, t 0, n 1, 1 i n.


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Finally, we present two exponential inequalities for independent Banach valued

random elements. See Sung [20, Lemma 1] for a proof of Lemma 2.5 (see also

[15]). The proof of Lemma 2.6 is also similar to that of Sung [20, Lemma 1].

Lemma 2.5. Let{Vi, 1 i n} be independent random elements such thatVi A a.s. for some constantA >0. Then for allt >0,

E

exp(tn

i=1

Vi)

exp

tEn

i=1

Vi + 2t2

ni=1

EVi2 exp(2tA)

.

Lemma 2.6. Let{Vi, 1 i n} be independent random elements such thatfor each1 i nVi Ai a.s. for some constantsA i> 0. Then for allt >0,

E

exp(tn

i=1

Vi)

exp

tEn

i=1

Vi + 4t2

ni=1

EVi2g(2tAi)

,

where g (x) =x2(ex 1 x).

3. ALMOSTSURECONVERGENCE

In this section, we establish the almost sure convergence of randomly weightedsums of independent random elements. The first theorem is a new result even when

Ani are constants. This theorem improves the cited result of Li et al. [16].

Theorem 3.1. Let p 1, and {Vi, i 1} be a sequence of independentrandom elements that are stochastically dominated by a random elementV with

EVp 2 and

(3.3)

ni=1

E(|Ani|2) = o(n2/p log1 n),

then


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(3.4) limn

1

n1/p

ni=1

AniVi= 0 a.s.

(2) If1 p 2 and

(3.5)

ni=1

E(|Ani|p) =O(n) for some0 < 0, we choose a positive integerN(to be specialized later), and

let

Xni =ViI(AniVi n1/p

log1

n),

Yni =ViI(n1/p log1 n < AniVi n

1/p

N),

and

Zni= ViI(AniVi > n1/p

N).

Proof of part(1) . For all t >0, by Lemma 2.5

(3.7)

P(n

i=1

AniXni> n1/p)

exp(tn1/p)E exp(tn

i=1

AniXni) (by the Markov inequality) exp(tn1/p)

exp

tEn

i=1

AniXni + 2t2

ni=1

EAniXni2 exp(2tn1/p log1 n)

.

Since EVp < , p >2, we have from (3.1) and (3.3) that

(3.8)

ni=1

EAniXni2 C

ni=1

E|Ani|2 =o(n2/p log1 n).


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Putting t = 2n1/p log n

, it follows from (3.2), (3.7) and (3.8) that there exists

1< n1/p) C

1

nand for all large n.

Therefore

(3.10)

n=1

P(n

i=1

AniXni > n1/p)< .

Now, we use the method in [12]. From the definition ofYni,

(3.11)

P(n

i=1

AniYni > n1/p)

P(there are at leastN suchis that AniVi> n1/p log1 n)

1i1 n1/p log1 n

n

i=1P(AniVi > n

1/p log1 n)

N

n

i=1

EAniVip

n logp n

N

Kp2n

i=1

E|Ani|2EVi

p

n logp n

N

Cn

i=1

E|Ani|2

n logp n

N C

logp nn12/p

N.

Choose Nlarge enough such that (1 2/p)N >1, we get from (3.11) that

(3.12)

n=1

P(n

i=1

AniYni> n1/p)< .

Now, we note that EVp < implies

i=1 P(Vi > i1/p) < , so by the

Borel-Cantelli lemma,

(3.13)

ni=1

Zni= O(1) a.s.

It follows from (3.1) and (3.13) that


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Putting t = 3n1/p log n

, it follows from (3.17) and the second part of (3.6)

that

(3.18) P(n

i=1

AniUi> n1/p) C

1

n2and for all large n.

Therefore

(3.19)

n=1P(

n

i=1AniUi > n

1/p)< .

By the Borel-Cantelli lemma,

(3.20) n1/pn

i=1

AniUi 0 a.s. as n .

By the argument similar to that of (3.13), we haven

i=1 Wiis bounded a.s. Thisimplies

(3.21) n1/pn

i=1

AniWi K

n1/p log n

ni=1

Wi 0 a.s. as n .

Conclusion (3.4) follows from (3.20) and (3.21).

Remark 3.2. We note:

(i) When Ani ani a.s., where {ani, n 1, 1 i n} is a bounded arrayof constants, then both condition (3.2) and the second part of (3.6) can be

replaced by

(3.22) n1/pn

i=1

aniVi 0 in probability asn .

The proof is standard (see Remark 4.3).

(ii) When p > 2, the condition (1.12) of Li et al. [16] is stronger than (3.3).So, Theorem 3.1 improves the cited result of Li et al. [16] even when the

non-randomly weighted case.

(iii) We now compare Part (1) of Theorem 3.1 with result of Sung [20] citedin Section (when p > 2). When Ani ani n

1/pbni, (3.1), (3.3) and

(3.22) coincide with (1.8), (1.9), (1.10), respectively. Sung [20] obtained the

complete convergence ofn

i=1 bniVni under condition EV2p < while

in Theorem 3.1, we obtain the almost sure convergencen

i=1 bniVi underconditionEVp < .


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(iv) Condition (3.5) is strong, but as we can see in Part (3), when (3.1) is strength-

ened tomax1in |Ani| K/ log n a.s., (3.5) can be removed. According toa result by Cuzick [5, Corollary 2.1 and Lemma 2.1], Part (3) does not hold

ifmax1in |Ani| K/ log n a.s. is weakened to (3.1).

The proof of the following theorem is similar to that of Part (1) of Theorem 3.1.

Theorem 3.3. Let p > 2, and {Vi, i 1} be a sequence of independentrandom elements that are stochastically dominated by a random elementV withEVp 1/2, 1 p < 2, {Vni, n 1, 1 i n} be an arrayof row-wise independent random elements that are stochastically dominated by a

random elementV with EVp < . Let{Ani, n 1, 1 i n} be an arrayof row-wise independent random variables. Assume that for all n 1, the randomvariables{Ani, 1 i n} are independent of{Vni,1 i n}. If

(4.1)

ni=1

E(|Ani|2) =O(n),

and

(4.2) limn

nE n

i=1

AniVniI(Vni n)= 0,

then

(4.3)

n=1

np2P n

i=1

AniVni> n< for all > 0.


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Proof. Noting that (4.3) is automatic ifp 0,

(4.4)

P n

i=1

AniVni> n

P

max1in

Vni > n

+P

n

i=1

AniWni > n

.

To obtain (4.3), it remains to show that

(4.5)

n=1

np2P

max1in

Vni > n

< ,

and

(4.6)

n=1

np2P

n

i=1

AniWni > n

< .

First, we have

(4.7)

n=1

np2P max1in

Vni> n

n=1

np2n

i=1

P

Vni> n

C

n=1

np1P(V > n)

=C

n=1

np1

i=n

P(i < V (i + 1))

=C

i=1i

n=1np1P(i < V (i + 1))

C

i=1

ipP(i < V (i + 1))

CEVp < .

This proves (4.5). Note that it follows from (4.2) that for all largen,

(4.8) nE n

i=1

AniVniI(Vni n)< /2.

So, to prove (4.6), it suffice to show that


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(4.9)

n=1

np2P n

i=1

AniWni En

i=1

AniWni> n/2< .

We have

n=1

np2P n

i=1

AniWni En

i=1

AniWni> n/2

4

2

n=1

np22E

ni=1

AniWni En

i=1

AniWni2

(by Chebyshevs inequality)

C

n=1

np22n

i=1

EAniWni2 (by Lemma 2.1 with r = 2)

=C

n=1

np22n

i=1

E|Ani|2EWni

2 (sinceAni is independent ofVni)

C

n=1

np22 n

i=1

E|Ani|2

n2P(V > n) + EV I(V n)2

(by (2.21))

C

n=1

np12EV I(V n)2 +C

n=1

np1P(V> n) (by (4.1))

C

n=1

np12n

i=1

E

V2I

(i 1) < V i

+CEVp (by (4.7))

=C

i=1

n=i

np12E

V2I

(i 1) < V i

+CEVp

C

i=1

ip2E

V2I((i 1) < V i)

+CEVp

C

i=1

ipP(i 1) < V i+CEVp CEVp < .

This proves (4.9). The proof of the theorem is completed.

Remark 4.2.

(i) When > 1, condition (4.2) is automatically satisfied. To see this, for all

n 1,


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ni=1

E|Ani|2

n

i=1

(EA2ni)1/22

(by Jensens inequality)

nn

i=1

EA2ni.

So, it follows from (4.1) that

(4.10)

ni=1

E|Ani|= O(n).

Hence

0 limn

nE n

i=1

AniVniI(Vni n)

limn

nn

i=1

EAniVniI(Vni n)

limn

Cn n

i=1

E|Ani|

EV I(V n) +nP(V> n)

(by (2.21))

limn

Cn1EV (by (4.10))

= 0 (since >1 andEV< ).

(ii) WhenX is a real separable Hilbert space andE Vni 0, by similar method(using the identityE(

ni=1 Vni)

2 =n

i=1 EV2

ni instead of Lemma 2.1) we

can prove that Theorem 4.1 still holds without (4.2).

Remark 4.3. When Ani ani a.s., where{ani, n 1, 1 i n} is an arrayof constants satisfying

(4.11) supn1, i1

|ani| < ,

then the condition (4.1) is satisfied automatically. Moreover, the condition (4.2) canbe replaced by

(4.12) nn

i=1

aniVni 0 in probability asn .

To see this, by Lemma 2.2 and (4.12), we can assume that for eachn 1,{Vni, 1 i n} are symmetric random elements. By Remark 4.2, we need only prove (4.2)when0 < 1. We have


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0 limn

nE n

i=1

aniVniI(Vni > n)

limn

nn

i=1

EaniVniI(Vni> n)

limn

Cn n

i=1

|ani|

EV I(V > n) (by (2.22))

limn

Cn1EV I(V> n) (by (4.11))

limn

CEV1/I(V> n) lim

nC

E

VpI(V > n)1/p

(by Jensens inequality and noting that p 1)

= 0 (sinceEVp < ).

It follows that

(4.13) nn

i=1

aniVniI(Vni > n) 0 in probability as n .

Combining (4.12) and (4.13), we have

(4.14) nn

i=1

aniVniI(Vni n) 0 in probability as n .

Now, (4.11) implies that there exists a constant Csuch thatnaniVniI(Vni n) |ani| Ca.s. for all n 1, 1 i n.Hence by Lemma 2.3 and (4.14), (4.2) follows.

Remark 4.4. Ifp 2, then Theorem 4.1 still hold if (4.1) is replaced by

(4.15)

ni=1

E(|Ani|q) =O(n) for some q >2(p 1)/(2 1).

The proof is similar to that of Theorem 4.1, by applying Lemma 2.1 with r = q/2.The verbatim argument is omitted.

Now, if we considerVni aniVni as a linear operator fni : X X, thenwe have the following theorem. The proof is exactly as that of Theorem 4.1 and

Remark 4.4.


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Theorem 4.5. Let > 1/2, p 1, {Vni, n 1, 1 i n} be an arrayof row-wise independent random elements that are stochastically dominated by a

random elementV withEVp < . Let{fni, n 1, 1 i n} be an array ofcontinuous linear operators fromX toX. If

(4.16) limn

nE n

i=1

fni(Vni)I(Vni n)= 0,

and if either

(4.17) 1 p < 2,n

i=1

fni2 =O(n),

or

(4.18) p 2,n

i=1

fniq =O(n) for some q >2(p 1)/(2 1),

then

(4.19)

n=1

np2P

ni=1

fni(Vni)

> n

< for all > 0.

By applying the fact that

E max1kn

ki=1

Vni

ni=1

EVni for all arbitrary array {Vni, n 1, 1 i n},

we obtain the following theorem. No row-wise independence conditions are needed.

Theorem 4.6. Let > 0, 0 < p < 1, {Vni, n 1, 1 i n} be an arrayof random elements that are stochastically dominated by a random elementV with

EVp < . Let {Ani, n 1, 1 i n} be an array of random variables.

Assume that for all n 1, 1 i n, the random variable A ni is independent ofVni. If

(4.20)

ni=1

E(|Ani|) =O(n),

then

(4.21)

n=1

np2P

max1kn

ki=1

AniVni> n< for all >0.


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Proof. Let Wni be as in the proof of Theorem 4.1. We need only prove the

theorem forp 1. For all >0,

(4.22)

n=1

np2P

max1kn

k

i=1

AniWni > n

1

n=1

np2E

max1kn

k

i=1

AniWni

(by Chebyshevs inequality)

Cn=1

np2n

i=1

E

AniWni

=Cn=1

np2n

i=1

E|Ani|EWni (sinceAni is independent ofVni)

Cn=1

np2 n

i=1

E|Ani|

nP(V> n)+EV I(V n)

(by (2.21))

Cn=1

np12EV I(V n)2 +Cn=1

np1P(V > n)

(by (4.20))

C

n=1np1

n

i=1 EVI(i 1)

< V i

+ CEVp=C

i=1

n=i

np1E

VI

(i 1) < V i

+C EVp

Ci=1

ipE

VI((i 1) < V i)

+CEVp

Ci=1

ipP

(i 1) < V i

+CEVp

CEVp < .

Similar to (4.7), we also have

(4.23)

n=1 np2Pmax1in Vni > n CEVp < .Since

P

max1kn

ki=1

AniVni> n

P

max1in

Vni> n

+ P

max1kn

k

i=1

AniWni > n

,

the conclusion (4.21) follows from (4.22) and (4.23).


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5. COROLLARIES AND EXAMPLES

In this section, we present several corollaries and examples. The first corollary

extends the result of Wang et al. [25] by including larger range for andp.

Corollary 5.1. Let > 1/2, p 1, {Vni, n 1, 1 i n} be an arrayof row-wise independent random elements that are stochastically dominated by a

random elementV withEVp < . Let{ani, n 1, 1 i n} be an array ofconstants satisfying

(5.1) supn1

sup1in

|ani| < .

If

(5.2) n n

i=1

aniVni

0 in probability as n ,then

(5.3)

n=1

np2P n

i=1

aniVni> n< for all > 0.

Proof. SetAni ania.s. Since (4.1) and (4.15) are immediately consequences

of (5.1), Corollary 5.1 follows from Theorem 4.1, Remark 4.3 and Remark 4.4

Remark 5.2. Corollary 5.1 extends the result of Wang et al. [25] and provides

better convergence rates.

(i) We only assume (5.2), whereas Wang et al. [25] assume (1.6).

(ii) We allow p 1, whereas Wang et al. [25] require thatp >1.

Corollary 5.3. Let1/2 t n1/t< for all >0.


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Proof. Letp= 2t, = 2/p= 1/t. Noting that1/2 t 1/2.Therefore Corollary 5.3 follows from Corollary 5.1.

Remark 5.4. Letani n1/tbni. Then Corollary 5.3 implies the p 0,

then

(5.10)

ni=1

E|Ani|r =O(n) for all 0 < r < q.


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The following example shows that in Theorem 4.1, we cannot replace (4.1) by the

weaker condition

(5.11)

ni=1

E|Ani|p =O(n).

Example 5.6. Let1 p n=P

|Vnn|>

= 1/ log(n + 1) for all 0 < < 1,

conclusion (4.3) fails.

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Le Van Thanh

Department of Mathematics

Vinh University

Nghe An 42118

Vietnam

E-mail: [email protected]

G. Yin

Department of Mathematics

Wayne State University

Detroit, MI 48202

U.S.A.

E-mail: [email protected]


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almost sure and complete convergence of randomly weighted sums of independent random elements in...

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