Stochastic comparison of multivariate random sums

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1 Stochastic comparison of multivariate random sums Rafa l Kuli Wroc law University Mathematical Institute Pl. Grunwaldzi 2/ Wroc law ruli@math.uni.wroc.pl April 13, 2004 Mathematics Subject Classification: 60E15 Key words and phrases: multivariate random sums, multivariate stochastic orders, convex order, directionally convex order, supermodular function Abstract We establish preservation results for the stochastic comparison of multivariate random sums of stationary, not necessary independent, sequences of nonnegative random variables. We consider convextype orderings, i.e. convex, coordinatewise convex, upper orthant convex and directionally convex orderings. Our theorems generalize the well-nown results for the stochastic ordering of random sums of independent random variables. 1 Introduction In this paper we establish preservation results for the stochastic comparison of multivariate random sums of nonnegative random variables. We consider convex, coordinatewise convex, upper orthant convex and directionally convex orderings. The first three are variability orderings, whereas the latter can be considered as variability-dependence orderings. We refer the reader to the boos [7] or [10]. Using so called supermodular functions we can extend the nown results [2], [8], [9] in the case of independent random variables to stationary sequences. There is a large number of applied sciences where random sums are used and comparison of them plays an important role in the stochastic models, 1

2 where the exact calculations of some quantities are not possible. For example in the context of multivariate shoc models preservation of stochastic ordering for multivariate random sums was established in [8] or [12]. In the actuarial science random sums were considered in [1], whereas in the context of queueing systems in [2] or in [4]. In the latter paper there was proved a preservation result in the directionally convex case not only for random sums but for more general functionals on stationary point processes. The paper is organized as follows. In section 2 we collect some needed definitions and technical lemmas. In section 3 we prove regularity properties of sums of random variables and then we apply it to the comparison results. In section 4 we point out some possibilities for obtaining assumptions in main theorems, i.e. we present examples of vectors of integer-valued random variables which can be compared in the above mentioned stochastic orderings. 2 Preliminaries We shall consider stationary sequences of nonnegative random variables. Precisely, we say that sequences {U i n} n 1, i = 1,..., are jointly stationary if for all n i 1, i = 1,...,, m 1. U 1 1,..., U 1 n 1,..., U 1,..., U n d = U 1 1+m,..., U 1 n 1 +m,..., U 1+m,..., U n +m, where d = denotes equality in distribution. We shall use some special classes of functions. A function ϕ : IR IR is convex on IR n if for all u = u 1,..., u, v = v 1,..., v and α 0, 1 we have ϕαu + 1 αv αϕu + 1 αϕv. For the convexity on IN = 1 this condition can be written as ϕn ϕn 1 2ϕn for all n 1. Define for 1 i, ɛ > 0 and arbitrary function ϕ : IR IR the difference operator ɛ i by ɛ iϕu 1,..., u = ϕu 1,..., u i 1, u i + ɛ, u i+1,..., u ϕ u 1,..., u for given u 1,..., u. A function ϕ : IR IR is called i coordinatewise convex if for all 1 i and ɛ i, ɛ j > 0 for all u = u 1,..., u ; ɛ i i ɛ j i ϕu 0 ii supermodular if for all 1 i < j and ɛ i, ɛ j > 0 for all u = u 1,..., u ; ɛ i i ɛ j j ϕu 0 2

3 iii directionally convex if it is supermodular and convex w.r.t. each coordinate or, equivalently ɛ i i ɛ j j ϕu 0 for all u = u 1,..., u and 1 i j. We shall say that the function is coordinatewise convex supermodular, directionally convex on IN if the above definitions are valid for u = u 1,..., u IN and ɛ i, ɛ j = 1. Moreover, we call a function ϕ : IR IR upper orthant lower orthant, convex upper orthant type if it has the form ϕu 1,..., u = h i u i, where h i : IR IR are increasing decreasing, increasing and convex. We denote by L cx L icx, L ccx, L iccx, L sm, L dcx, L idcx, L uo, L lo, L uo cx the class of convex increasing and convex, coordinatewise convex, increasing and coordinatewise convex, supermodular, directionally convex, increasing and directionally convex, upper orthant type, lower orthant type, convex upper orthant type functions. Here, we mean increasing or decreasing in the wea sense. Recall that the following relations hold: L idcx L dcx L sm, L dcx L ccx. Note that a function which is directionally convex need not to be convex and a function which is convex need not to be directionally convex. As an example consider ϕu 1,..., u = maxu 1,..., u, which is convex, but not directionally convex. Example 2.1 We present some examples of functions which are members of the above classes. Let u i 0, i = 1,...,. 1. ϕu 1,..., u = f u i L dcx for convex f : IR IR + ; 2. ϕu 1,..., u = f u i L dcx for increasing and convex f : IR IR + ; 3. ϕu 1,..., u = fu i L dcx for convex f : IR IR + ; 4. ϕu 1,..., u = fu i L dcx for increasing convex f : IR IR + ; The above classes of functions generate stochastic orderings. For random vectors U = U 1,..., U, Ũ = Ũ1,..., Ũ we write U < a Ũ if IE[fU] IE[fŨ] for all f from a given class L a for which expectations exist. We shall need the following technical lemma. 3

4 Lemma 2.2 Let u = u 1,..., u. i Let f : IR IR L cx. Then φ : IR IR defined as φu = f u i is supermodular and directionally convex on IR +. ii Let ϕ : IR IR L dcx and u j i 0, 1 j, 1 i n j. Then φ defined as n1 φu 1 1,..., u 1 n 1,..., u 1,..., u n = ϕ u 1 n i,..., is directionally convex on IR n 1+ +n +. u i Proof. The i result is taen from [6, p. 152]. In order to obtain ii result we proceed as follows. Let n = n n. We need to show that for all 1 i j n and ɛ i, ɛ j > 0, ɛ i i ɛ j j φ 0. Observe that for 1 l j n 1 ɛ l l ɛ j j ψu 1 1,..., u 1 n 1,..., u 1,..., u n = ϕ n1 n u 1 i + ɛ l + ɛ j,..., u i n1 +ϕ u 1 n i,..., u i n1 ϕ u 1 n i + ɛ l,..., u i n1 ϕ u 1 n i + ɛ j,..., u i 0 from the convexity of ϕ w.r.t. first coordinate. Similarly for n 1 + +n r 1 < l n n r, n n s 1 < j n n s, r < s ɛ l l ɛ j j ψu 1 1,..., u 1 n 1,..., u 1,..., u n r n = ϕ..., u r n s i + ɛ l,..., u s i + ɛ j,... from supermodularity of ϕ. 0 +ϕ ϕ ϕ...,...,..., n r n r n r u r i,..., n s u s i,... u r n s i + ɛ l,..., u s i,... u r n s i,..., u s i + ɛ j,... 4

5 3 Main results We start with some technical lemmas. The first one is a straightforward consequence of the definition of supermodular functions cf. [1]. Lemma 3.1 Let {Un} i n 1, i = 1,..., be sequences of nonnegative random variables. Then for all functions ϕ L sm [ n1 ] ψn 1,..., n = IE ϕ 1 n,..., is supermodular on IN. Lemma 3.2 Let {Un} i n 1, i = 1,..., be stationary sequences of nonnegative random variables such that for all i j, {Un} i n 1 is independent of {Un} j n 1. Denote [ n1 ] ψn 1,..., n = IE ϕ 1 n,...,. i If ϕ L dcx L idcx then ψ L dcx L idcx on IN ; ii If ϕ L ccx L iccx then ψ L ccx L iccx on IN ; iii If ϕ L uo cx then ψ L uo cx on IN. Proof. i Let ϕ L dcx. First, we can apply Lemma 3.1 in order to obtain that ψ is supermodular. We need only to show that ψ is convex w.r.t. n i, i = 1,...,, i.e. it is coordinatewise convex. Let U i n i = U i 1,..., U i n i and write f n u 1,..., u n = n u i. Then by independence assumption ψn 1 + 1, n 2,..., n + ψn 1 1, n 2,..., n 2ψn 1,..., n = = IE [ ϕ f n 1+1 U 1 n 1 +1, f n 2 U 2 n 2,..., f n U n ] +IE [ ϕ f n 1 1 U 1 n 1 1, f n 2 U 2 n 2,..., f n U n ] 2IE [ ϕ f n 1 U 1 n 1, f n 2 U 2 n 2,..., f n U n ] = IE [ ϕ f n 1+1 U 1 n 1 +1, f n 2 U 2 n 2,..., f n U n ] +IE [ ϕ f n 1+1 0, U 1 1,..., U 1 n 1 1, 0, f n 2 U 2 n 2,..., f n U n ] IE [ ϕ f n 1+1 0, U 1 1,..., U 1 n 1, f n 2 U 2 n 2,..., f n U n ] IE [ ϕ f n 1+1 U 1 1,..., U 1 n 1, 0, f n 2 U 2 n 2,..., f n U n ] = IE [ ϕ f n 1+1 U 1 n 1 +1, f n 2 U 2 n 2,..., f n U n ] +IE [ ϕ f n 1+1 0, U 1 2,..., U 1 n 1, 0, f n 2 U 2 n 2,..., f n U n ] IE [ ϕ f n 1+1 0, U 1 2,..., U 1 n 1 +1, f n 2 U 2 n 2,..., f n U n ] IE [ ϕ f n 1+1 U 1 1,..., U 1 n 1, 0, f n 2 2 U 2 n 2,..., f n U n ]. 5

6 In the second equality we used symmetry property of f n, whereas in the third we used stationarity. Write the above equation in the form ψn 1 + 1, n 2,..., n + ψn 1 1, n 2,..., n 2ψn 1,..., n = = IE [ ϕ f n1+1 U 1 n 1 +1, f n2 u 2 n 2,..., f n u n ] dip U1 u 1 + IE [ ϕ f n1+1 0, U2 1,..., Un 1 1, 0, f n2 u 2 n 2,..., f n u n ] dip U1 u 1 IE [ ϕ f n1+1 0, U2 1,..., Un , f n2 u 2 n 2,..., f n u n ] dip U1 u 1 IE [ ϕ f n1+1 U1 1,..., Un 1 1, 0, f n2 u 2 n 2,..., f n u n ] dip U1 u 1 Here IP U1 denotes the distribution of U 2 n 2,..., U n and u 1 u 2 n 2,..., u n. Because for all n 1, f n is linear and ϕ is convex w.r.t. the first coordinate because it is directionally convex we obtain that ϕ f n 1+1 u 1 1,..., u 1 n 1, u 1 n 1 +1, f n 2 u 2 n 2,..., f n u n is supermodular w.r.t. u 1 1,..., u 1 n 1 +1 Lemma 2.2i. Therefore ψn 1 + 1, n 2,..., n + ψn 1 1, n 2,..., n 2ψn 1,..., n 0 which ends the proof. ii The proof is similar to i. iii Let ϕ L uo cx. Then ψn 1,..., n can be written in the form nj ψn 1,..., n = IE h j U j i, j=1 where h j are increasing and convex. From independence we have nj ] ψn 1,..., n = IE [h j U j i. j=1 Now, using i with = 1, g j n j = IE [ h j nj U j i ], j = 1,...,, are increasing and convex. Therefore ψn 1,..., n can be written as the product of increasing and convex functions, which ends the proof. In all cases increasingness is obvious. Using the similar argument as above with Lemma 2.2ii we can prove the following lemma. 6

7 Lemma 3.3 Let {Un} i n 1, i = 1,..., be jointly stationary sequences of nonnegative random variables. Then for all functions ϕ L dcx L idcx [ n ] n ψn = IE ϕ 1,..., is convex increasing and convex on IN. Remar 3.4 All the above results in the case = 1 mean that ψn = ϕ n U i is convex for convex ϕ. This result was proved in [9, p. 278], in the case of {U n } n 1 iid nonnegative random variables. In [2] it was shown in the case of {U n } n 1 - nonstationary sequences of independent nonnegative random variables such that IE[hU n ] IE[hU n+1 ] for all h L cx h L icx. Maowsi and Phillips [5] showed that for iid nonnegative random variables {U n } n 1, the function ψn/n is increasing. From the above lemmas we have directly the following theorems. Let N = N 1,..., N, Ñ = Ñ1,..., Ñ be vectors of nonnegative, integer-valued random variables and N 1 T = 1 N,...,, T = Ñ 1 U 1 i,..., Ñ. Theorem 3.5 Under assumptions of lemma 3.2: i If N < dcx < idcx Ñ then T < dcx < idcx T; ii If N < ccx < iccx Ñ then T < ccx < iccx T; iii If N < uo cx Ñ then T < uo cx T. Let now N, Ñ be nonnegative and integer-valued random variables. Theorem 3.6 Under assumptions of lemma 3.3, if N < cx < icx Ñ then N N Ñ Ñ 1,..., < dcx < idcx 1,...,. Pellerey [8] showed the following preservations results which are parallel to the Theorem 3.5. Assume that {U i n} n 1, i = 1,..., are independent sequences of nonnegative independent random variables such that for each f L cx f L icx, IE[fU i n] IE[fU i n+1], n 1, i = 1,...,. Then N < ccx Ñ implies T < ccx T N < icx < iccx, < uo cx Ñ implies T < icx < iccx, < uo cx T. Moreover, without any independence assumption, N < uo < lo Ñ implies T < uo < lo T. In [1] it was shown under the same assumptions that N < sm Ñ implies T < sm T. 7

8 4 Criteria for variability orderings In this section we point out some possibilities for obtaining sufficient conditions of Theorem 3.5, i.e. we present examples of integer valued random variables N 1,..., N, Ñ1,..., Ñ such that N < a Ñ, where < a is one of the above orderings. i Assume that for integer-valued random variables we have N < icx Ñ. For example we can tae N := Nt, t 0, where Nt, t 0 is a stationary counting process Nt := n=0 IIT n t for which interpoint distances X n = T n T n 1, n 1, have distribution F X and the first distance T 0 from the origin has distribution FXx r := 1 x 0 1 F Xudu. We define in the similar way elements with E[X 1 ] tilde. If for all n 1 n n X i < cx X i 1 then for all t 0, Nt < cx Ñt cf. [3]. For instance, 1 holds if X 1,..., X n < dcx X 1,..., X n for all n 1, i.e. the sequence of interpoint distances in the second counting process is more dependent and more variable than in the first one. We refer to [3] for examples. In the case of renewal counting processes we can tae as X n and X n any positive random variables which are ordered w.r.t. < cx Note that < cx is closed under convolution. Another possibility to obtain N < cx Ñ is the following. Assume that N and Ñ have finite support. Then N < cx Ñ if and only if probability function of Ñ can be obtained from N by a finite sequence of local mean preserving spreads which, roughly speaing, means that the mass is removed from a point x in the support of N and shifted to y and z y < x < z in the support of Ñ, but the mean remains constant See [7], Definition and Theorem for more details. Let now {V n } n 1 be stationary sequence of {1,..., }-valued random variables. Define for j = 1,..., and N N j = IIV i = j Ñ Ñ j = IIV i = j. Then using Theorem 3.6 with U j i = IIV i = j we have N < idcx Ñ. ii Consider a vector N = N 1,..., N of integer-valued random variables with the same marginal distribution F. It follows from Lorentz inequality [11] that N 1,..., N < dcx N 1,..., N 1. 8

9 iii Assume that N i < cx Ñ i, i = 1,...,. If N and Ñ are comonotone random vectors or have a common conditionally increasing copula then N < idcx Ñ See [7, Lemma , Theorem ] for more details. References [1] M. Denuit, C. Genest and E. Marceau, Criteria for the Stochastic Ordering of Random Sums, with Actuarial Applications, Scand. Actuar. J. 2002, [2] A. Jean-Marie and Z. Liu, Stochastic comparison for queueing models via random sums and intervals, Adv. Appl. Prob , [3] R. Kuli and R. Szeli, Sufficient conditions for long range dependence of stationary point processes on the real line, J. Appl. Prob. 38, [4] R. Kuli and R. Szeli, Dependence orderings for some functionals of multivariate point processes, submitted. [5] A. Maowsi and T. Philips, Stochastic convexity of sums of i.i.d. nonnegative random variables with applications, J. Appl. Prob , [6] A. W. Marshall and I. Olin, Inequalities: Theory of Majorization and Its Applications, Academic Press, [7] A. Müller and D. Stoyan, Comparison Methods for Stochastic Models and Riss, Wiley, New Yor, [8] F. Pellerey, Stochastic Comparisons for Multivariate Shoc Models, J. Multivariate Anal , [9] S. Ross, Stochastic Processes, Wiley, New Yor, [10] M. Shaed and J.G. Shanthiumar, Stochastic Orders and Their Applications, Academic Press, London, [11] M. Shaed and J.G. Shanthiumar, Supermodular Stochastic Orders and Positive Dependence of Random Vectors, J. Multivariate Anal , [12] T. Wong, Preservation of multivariate stochastic orders under multivariate Poisson shoc models, J. Appl. Prob ,