CRRAO Advanced Institute of Mathematics, Statistics and Computer Science (AIMSCS)

Author (s): C.R. Rao, D.N. Shanbhag

Title of the Report: Approaches to Damage Models and related results in Applied Probability

Research Report No.: RR2014-14

Date: May 19, 2014

Prof. C R Rao Road, University of Hyderabad Campus, Gachibowli, Hyderabad-500046, INDIA.

www.crraoaimscs.org

Research Report

Approaches to Damage Models and related

results in Applied Probability

C. Radhakrishna Rao

Hon. Advisor, CRRAO AIMSCS,

Research Professor, University at Buffalo

and

Damodar N. Shanbhag

Sheffield, England, UK.

AbstractApproaches based on non-negative matrices or, in particular, on

specialized versions of de Finettis theorem, have led to some resultsof relevance to damage models, including those on Integrated CauchyFunctional Equation and Extended Spitzer Integral RepresentationTheorem. Now we revisit these results shedding further light on someof their aspects; in the process of doing this, we observe, amongstother things, that the latter of the two results referred to here has alink with the Weyl integral from fractional calculus.

Keywords: Raos damage model, Lau-Rao-Shanhag theorems, Spitzer inte-gral representation theorem, Rao-Rubin-Shanbhag theorems, Non-negativematrices, Fractional calculus, Weyl integral, Denys theorem, de Finettistheorem, Branching processes, Markov processes.

1 Introduction

The notion of damage models was introduced, giving some motivation andrelevant supporting material, by Rao (1963). This has generated consider-able interest amongst researchers specializing in characteristic properties ofstochastic models and related integral equations.

Email: crr1@psu.eduCorresponding author. Email: damodarshanbhag@hotmail.com, Home Address: D.N.

Shanbhag, 3 Worcester Close, Sheffield S10 4JF, England, U.K.

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In mathematical terms, a damage model defined by Rao may be viewed asa random vector (X,Y) with non-negative integer-valued components X,Ysuch that Y X almost surely with 0 < P{Y = X} 1; in the context ofa damage model, the conditional distribution of Y given X, i.e. any chosenversion of it, is usually referred to as the survival distribution of the model,and the following as the Rao-Rubin (RR(0)) condition:

P{Y = y} = P{Y = y|XY = 0}, y = 0, 1, . . . . (1.1)

The multivariate versions of the terminologies and certain of their vari-ations have also appeared in the literature; see e.g. Chapter 7 of Rao andShanbhag (1994) or Rao and Shanbhag (2004).

Amongst various important results that one comes across on damagemodels, there are those that are due to Rao and Rubin (1964) and Shanbhag(1977); in particular, from what is observed in the latter article, it followsthat under certain conditions, the problem of identifying the solution to (1.1)reduces to that of solving a discrete version of an integral equation, studied,terming it as the Integrated Cauchy Functional Equation (ICFE), in Lau andRao (1982). Also, it may be noted here that more general versions of ICFEhave essentially been studied in Chapter 3 of Rao and Shanbhag (1994) andin other places such as, Shanbhag (1991).

The main result of Shanbhag (1977) subsumes several specialized results,including that of Rao and Rubin (1964); Chapter 7 of Rao and Shanbhag(1994) highlights various important results appearing in the literature ondamage models. Included in these, besides the result of Shanbhag and certainof its extensions and variations, is an extended version of Spitzers integralrepresentation theorem relative to stationary measures of certain discretebranching processes, with a link to damage models. More recently, Rao andShanbhag (2004) have shown that this latter result holds even when one ofthe moment assumptions in it, i.e. in the notation used in the literature,that m is finite, is dropped.

Rao and Shanbhag [(1994, Section 4.4), (1998), (2004)] and Rao et al.(2002) have shown explicitly or otherwise that there exist approaches todamage models based on non-negative matrices or exchangeability, involvingamongst other things, certain special cases of de Finnetis theorem. Chap-ters 2 and 3 of Rao and Shanbhag (1994) provide us with further informationon some of the research material met in the references such as Alzaid et al.(1987b), Rao and Shanbhag (1991) and Shanbhag (1991), involving ideasbased on exchangeability or, in particular, de Finettis theorem, to solve cer-tain versions of Choquet-Deny (1960) and Deny (1961) equations, or theirvariations. Also, from the cited literature, it is now evident that we have

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proofs based on versions of de Finettis theorem or ICFE for certain poten-tial theoretic results such as Hausdorffs theorem on completely monotonesequences, and Bernsteins and Bochners theorems on completely monotonefunctions (and hence also, their versions on absolutely monotone functions).

In the paper, we make some new observations on integral equations, ofrelevance to damage models, involving partially aspects of non-negative ma-trices or exchangeability, and show, in particular, that the Weyl integral metin fractional calculus has a link with the extended Spitzer integral represen-tation theorem referred to above. We also highlight in this paper some ofthe major implications of these findings.

2 Generalized Discrete ICFE with applica-

tion to damage models

Before dealing with the main findings on the generalized discrete ICFE in thissection, we shall briefly revisit an application of such equations to damagemodels, met essentially in Rao and Shanbhag (2004, pp. 67-68):

Let S be a countable Abelian semigroup with zero element, equipped withdiscrete topology, and v and w be non-negative real-valued functions definedon S, with v satisfying additionally that v(0) > 0. Note that there are casesof (S, v.w), in which

v(x) =yS

v(x + y)w(y), x S. (2.1)

Given (S, w), possibly, meeting some additional conditions, the problemof identifying the (class of) functions v, for which (2.1) is valid, may be viewedas that of solving a version of discrete ICFE. Suppose S S (not dependingon w) is such that, in the case of supp(w)(= {x : w(x) > 0}) S, (2.1)is met if and only if (iff, for short) there exists a family {e(x, .) : x S} ofnon-negative random variables defined on a probability space, meeting therequirements that e(x, .)e(y, .) = e(x + y, .), x, y S, xS e(x, .)w(x) = 1and E(e(x, .)) = v(x)

v(0), x S. (There is obviously no loss of generality if we

consider in (2.1), in place of v, its normalised version with v(0) = 1.)Assume now that a : S (0,) and b : S [0,) are such that

b(0) > 0 and there exists c : S (0,) as the convolution of a and b, andY and Z are random elements defined on a probablility space, with valuesin S, such that

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P{Y = y, Z = z} = g(y + z)a(y)b(z)c(y + z)

, y, z S,

where {g(x) : x S} is a probability distribution. If supp(b) S, then iteasily follows that

P{Y = y} = P{Y = y|Z = 0}, y S,iff g(x)

c(x) E(e(x, .)), x S, where {e(x, .)} meets the requirements of

{e(x, .)}, referred to above, but, with, for some > 0, .b appearing inplace of w.(Note that by assumption, c(x) a(x)b(0) > 0 for each x Sand c exists at least in the cases with a, b bounded and

xS a(x) < or

xS b(x) < .)Let S(w) be the smallest subsemigroup of S containing {0} supp(w).

Then, if S = S(w), the existence of S, with stated property, is implied bywhat is observed as an application of a result on non-negative matrices inRao and Shanbhag (1994, pp. 98-99) or by an argument based on a versionof deFinettis theorem produced in Rao and Shanbhag (1998, Section 2) toobtain, essentially, a criterion for the validity of (2.1).

The information that we have gathered above tells us, in particular, thatthe following theorem holds; for some results of relevance to this theorem,see, also, Ressel (1985) and Rao et al. (2002).

Theorem 2.1: Let S be as defined earlier. Also, let v : S [0,) with,v(0) = 1, w : S [0,) and S(w) be as defined above. Then

v(x) =yS

v(x + y)w(y), x S(w), (2.2)

iff there exists a family {e(x, .) : x S(w)} of non-negative random variablesdefined on a probability space, satisfying e(x, .)e(y, .) = e(x + y, .), x, y S(w),

xS(w) e(x, .)w(x) = 1 and E(e(x, .)) = v(x), x S(w).

(We use here the notation e again for simplicity, but in a different context.)

Remark 2.1: Theorem 2.2 of Rao and Shanbhag (2004) proved via a simpleversion of deFinettis theorem, is indeed a corollary to Theorem 2.1 appearingabove in which S = S(w) = ({0, 1, 2...})k (k being a positive integer). Thecited article obtains (the multivariate versions of) Hausdorffs theorem oncompletely monotone sequences and Rao-Rubin-Shanbhag theorems on dam-age models, as obvious corollaries to this result; see, also, Rao and Shanbhag(1994, pp. 166-167) for some relevant findings on damage models.

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Remark 2.2: We can shed further light on the link between Hausdorffs the-orem and the specialized version of Theorem 2.1, referred to in the previousremark. In the case of S = S(w) = ({0, 1, 2...})k, w of Theorem 2.1 is suchthat w(x) > 0 for each x of length 1, enabling us to define (v, w), where(in obvious notation) for each x(= (x1, x2, .., xk)), v

(x) = v(x)k

r=1 wxrr

and w(x) = w(x)k

r=1 wxrr with wr = w(r1, ..., rk), r = 1, .., k, in-

volving the standard notation of Kronecker delta. Suppose in the case ofS = S(w) = ({0, 1, 2...})k, (v, w) satisfies (2.2), then, in the notation ofRao and Shanbhag (1994, p.75), for each (n1, n2, .., nk) ({0, 1, 2...})k, onecan observe inductively (with respect to (m1,