Monotonicity and Aging Properties of Random Sums

Monotonicity and Aging Properties of Random Sums Jun Cai and Gordon E. Willmot Department of Statistics and Actuarial Science University of Waterloo Waterloo, Ontario Canada N2L 3G1 E-mail: jcai@uwaterloo.ca, gewillmo@uwaterloo.ca Abstract In this paper, we discuss the distributional properties of random sums. We first derive conditions under which the distribution of a binomial sum is PF 2 and then show under the same conditions the distribution of a Poisson sum is PF 2 by approximating a Poisson sum by a sequence of binomial sums. The PF 2 property reveals the monotonicity property of the reversed failure rates of certain compound Poisson distributions. Further, we discuss a class of random sums and derive the NWUE aging property of random sums in the class. The result, together with existing results, shows that the aging properties of many random sums can be characterized uniquely by the aging properties of the primary distributions of the random sums whatever the underlying distributions of the random sums are. Key words: random sum, Poisson sum, binomial sum, geometric sum, discrete distribution, failure rate, reversed failure rate, decreasing failure rate, increasing failure rate, decreasing reversed failure rate, PF 2, NWUE. Acknowledgments: Support for the authors from the Natural Sciences and Engineering Research Council of Canada (NSERC) is acknowledged. Support for the second author from the Munich Reinsurance Company is also gratefully acknowledged. 1

1 Introduction Let {X i, i = 1, 2, } be a sequence of independent and identically distributed (i.i.d.) nonnegative random variables independent of a counting random variable N, which is a nonnegative integer-valued random variable. A random sum S N is a random variable defined by S N = X 1 + + X N and S N = 0 if N = 0. The distribution of a random sum is called a compound distribution. If X 1 is supported on {0, 1,...}, the random sum S N is called a discrete random sum. If X 1 is supported on [0, ), the random sum S N is called a continuous random sum. When we say a random sum, it includes both discrete and continuous cases. A random sum is an important probability model and appears in many fields of statistics and probability. A random sum itself has a simple definition. However, it is not easy to derive analytic properties of a random sum. Two of the important random sums are geometric and Poisson sums, in the former the counting random variable N is geometric random variable while in the latter N is a Poisson random variable. The distributions of the two random sums are called compound geometric and compound Poisson distributions, respectively. In particular, the class of discrete Poisson sums consists with the class of discrete infinitely divisible random variables, and the class of discrete geometric sums is a subclass of the class of Poisson sums. See, for example, Steutel and Van Harn (2004) and references therein. Some distributional properties of a geometric sum have been obtained in the literature. For example, Shanthikumar (1988) has proved that the failure rate of a discrete geometric sum X 1 + + X N is decreasing if X 1 has a decreasing failure rate, namely the DS-DFR property is preserved under geometric sum (see Section 2 for the definition of DS-DFR). Brown (1990) showed that a geometric sum S N = X 1 + + X N always has the NWU aging property whatever the distribution of X 1 is, namely if N is a geometric random variable, then Pr{S N > x + y S N > x} Pr{S N > y}, x 0, y 0 always holds whatever the distribution of X 1 is. Other distributional properties of a geometric sum can be found in Kováts and Móri (1992), Lin (2002), Willmot (2002), Willmot and Cai (2001, 2004), and references therein. Motivated by the monotonicity and aging properties of a geometric sum, we consider monotonicity and aging properties of a Poisson sum and other random sums in this paper. This paper is organized as follows: In Section 2, we first give the definitions of the classes of discrete distributions including DS-DFR (DS-IFR), D-DFR (D-IFR), D-PF 2, and D-LCVX (D-LCAV), and then summarize some preliminary results about these classes. 2

In Section 3, we discuss the PF 2, DFR, and IFR properties of binomial and Poisson sums. We show that a binomial or Poisson sum X 1 + + X N is PF 2 if X 1 has a decreasing probability mass function. The PF 2 property of the Poisson sum reveals the monotonicity property of the reversed failure rates of certain compound Poisson distributions. Further, we show that the DS-DFR property is not preserved under Poisson sum. For a continuous Poisson sum, we prove that a binomial or Poisson sum X 1 + + X N is PF 2 if X 1 has a decreasing density function on (0, ), which relaxes the strictly decreasing condition derived by Kijima and Nakagama (1991), who used an different argument. In Section 4, we discuss a class of random sums and derive the NWUE aging property of random sums in the class. The result, together with existing results, shows that the aging property of a random sum X 1 + + X N may be characterized uniquely by the aging property of the primary distribution of N whatever the underlying distribution of X 1 is. In this paper, we discuss both discrete and continuous random sums. Throughout this paper, the terms increasing and decreasing mean non-decreasing and non-increasing respectively. Moreover, F (x) = 1 F (x) denotes the survival function of a distribution function F (x). 2 PF 2 and other classes of discrete distributions Let a counting random variable X have the probability mass function p n = Pr{X = n}, n = 0, 1,. Then the failure rate of X is denoted by h n = Pr{X = n X n} = p n i=n p i = p n Q n, n = 0, 1,, where Q n = Pr{X n} = k=n p k. Furthermore, the reversed failure rate of X is defined by r n = Pr{X = n X n} = p n ni=0 p i = p n Q n, n = 0, 1,, where Q n = Pr{X n} = n k=0 p k. The failure rate is a well-known function in statistics and probability. However, the reversed failure rate has also appeared in many fields of statistics and probability. In particular, a reversed failure rate is called a retro-hazard rate in survival analysis and a reversed hazard rate in reliability and stochastic ordering, see, for example, Kalbfleish and Lawless (1989), Anderson et al (1993), Shaked and Shanthikumar (1994), Nanda and Shaked (2001), and references therein. Furthermore, some important classes of discrete distributions can be characterized by the monotonicity properties of failure rates and reversed failure rates. 3

Definition 2.1 Let E = {..., 1, 0, 1,...}. A non-negative function f(x, y) defined on E E is called TP 2 in x and y, if f(x 1, y 1 ) f(x 2, y 2 ) f(x 1, y 2 ) f(x 2, y 1 ) for all x 1 < x 2 and y 1 < y 2. A non-negative function g(x) defined on E is said to be PF 2 in x if g(x y) is TP 2 in x and y. Remark 2.1 A non-negative function g(x) defined on {0, 1, 2,...} is also viewed as a function defined on E using the convention g(n) = 0, n = 1, 2,... Thus, a non-negative function g(x) defined on {0, 1, 2,...} is PF 2 if and only if g(n + 1)/g(n) is decreasing in n = 0, 1, 2,... with the convention 0/0 = 0. Such a PF 2 sequence is also called a log-concave sequence. A detailed study of TP 2 and PF 2 functions can be found in Karlin (1968). Definition 2.2 A counting random variable X or its distribution is said to be DS-IFR, written X DS-IFR, if Qn = Pr{X n} is PF 2 in n, or equivalently, if Qn+1 / Q n is decreasing in n = 0, 1,... It follows from h n = 1 ( Q n+1 / Q n ), n = 0, 1, 2,... that X is DS-IFR if and only if the failure rate h n of X is increasing in n = 0, 1,... Definition 2.3 A counting random variable X or its distribution is said to be DS-DFR, written X DS-DFR, if the failure rate h n of X is decreasing in n = 0, 1,..., or equivalently, Q n+1 / Q n is increasing in n = 0, 1,... Definition 2.4 A counting random variable X or its distribution is said to be D-PF 2, written X D-PF 2, if Q n = Pr{X n} is PF 2 in n, or equivalently, Q n+1 /Q n is decreasing in n = 0, 1,... It follows from r n = 1 (Q n /Q n 1 ) 1, n = 1, 2,... that X is D-PF 2 if and only if the reversed failure rate r n of X is decreasing in n = 1, 2,... Definition 2.5 A counting random variable X or its distribution is said to be D-DFR (D- IFR), written X D-DFR (D-IFR), if the failure rate h n of X is decreasing (increasing) in n = 1, 2,... We point out that the monotonicity properties of the failure rate h n in D-DFR (D-IFR) and the reversed failure rate r n in PF 2 start from n = 1 while the monotonicity property of the failure rate h n in DS-DFR (DS-IFR) starts from n = 0, namely DS-DFR (DS-IFR) has a stronger monotonicity property than D-DFR (D-IFR). Indeed, DS-DFR D-DFR and DS-IFR D-IFR. 4

Definition 2.6 A counting random variable X or its probability mass function {p n, n = 0, 1,...} is said to be log-convex, written X D-LCVX, if {p n, n = 0, 1, } is log-convex, namely p 2 n+1 p n p n+2, n = 0, 1,. (2.1) If the reversed inequality in (2.1) holds, X or its probability mass function is said to be log-concave, written X D-LCAV. We note that X is D-LCVX (D-LCAV) if and only if p n+1 /p n is increasing (decreasing) in n = 0, 1,... It is also known that D-LCVX (D-LCAV) is a subclass of DS-DFR (DS-IFR), or D-LCVX DS-DFR and D-LCAV DS-IFR. The classes of DS-DFR (DS-IFR), D-DFR (D-IFR), and D-LCVX (D-LCAV) are wellknown classes of discrete distributions in reliability and applied probability. Also, the class of D-PF 2 is of interest in statistics and probability. We summarize the important properties of D-PF 2 in the following lemma: Lemma 2.1 (i) For a discrete distribution {p n, n = 0, 1,...}, if p n is decreasing in n = 1, 2,..., then the distribution {p n, n = 0, 1,...} is D-PF 2. (ii) D-DFR D-PF 2. (iii) D-LCAV D-PF 2. (iv) D-PF 2 is closed under convolution, namely if X 1,..., X n are independent D-PF 2 random variables, then X 1 + + X n is also D-PF 2. (v) D-PF 2 is closed under weak convergence, namely if X 1, X 2,... are a sequence of D- PF 2 random variables and X n convergences to X in distribution as n, then X is also D-PF 2. Proof. (i) We know that Q n = n k=0 p k is increasing in n = 1, 2,... and r n = p n /Q n for n = 1, 2,... Thus, if p n is decreasing in n = 1, 2,..., then r n is decreasing in n = 1, 2,... Hence, the distribution {p n, n = 0, 1,...} is D-PF 2. (ii) Let a discrete distribution {p n, n = 0, 1, } be D-DFR. Since Q n = k=n p k is decreasing in n = 0, 1, 2,, it follows from p n = h n Qn that p n is decreasing in n = 1, 2, if h n is decreasing in n = 1, 2,... Hence p n is decreasing in n = 1, 2, if {p n, n = 0, 1, } is D-DFR. Thus, (ii) follows from (i). 5

(iii) The proof follows from 1 r n = Q n p n = n i=0 p i p n = n i=0 n 1 k=i p k p k+1, n = 1, 2, and the fact that {p n, n = 0, 1, 2,...} is D-LCAV if and only if p n+1 /p n is decreasing in n = 0, 1, 2,... (iv) The result is given in Lemma 1.1 of Kijima (1992). (v) The result follows simply from the definition of a D-PF 2 distribution and the fact that X n convergences to X in distribution as n if and only if for any k = 0, 1, 2,..., lim Pr{X n = k} = Pr{X = k}. n Remark 2.2 It follows from the proof of Lemma 2.1 (ii) that if {p n, n = 0, 1, 2,...} is DS- DFR (D-DFR) then p n is decreasing in n = 0, 1, 2,...(n = 1, 2,...). Also, we point out that Lemma 1.1 of Kijima (1992) has showed DS-DFR D-PF 2. We illustrate the implications of these classes of discrete distributions as follows: D-LCVX DS-DFR D-DFR D-PF 2. D-LCAV D-PF 2. D-LCAV D-IFR D-PF 2. 3 The PF 2 property of binomial and Poisson sums In this section, we give conditions under which discrete binomial and Poisson sums are D- PF 2. Throughout this paper, we denote the equality in distribution of random variables X and Y by X = d Y. Lemma 3.1 Suppose that N, N 1,, N k are independent counting random variables independent of a sequence of i.i.d. non-negative random variables {X i, i 1}. If N = d N 1 + + N k, then the random sum S N = X 1 + + X N satisfies that S N = d S N1 + + S Nk, where S N1,, S Nk are independent random sums with S Nj = d X 1 + + X Nj, j = 1,, k. 6

Proof. The proof follows simply from the fact that the Laplace transforms of S N and S N1 + + S Nk are equal. First, we give conditions under which a discrete binomial sum is D-PF 2. Theorem 3.1 Let N be a binomial random variable and {X i, i 1} be a sequence of i.i.d. counting random variables independent of N. If the probability mass function Pr{X 1 = n} is decreasing in n = 1, 2,..., then the binomial sum S N = X 1 + + X N is D-PF 2. Proof. Denote the probability mass function of N by ( ) l Pr{N = k} = p k (1 p) l k, 0 < p < 1, k = 0, 1,, l. k Let ν 1,, ν l be i.i.d. Bernoulli random variables with Pr{ν 1 = 1} = 1 Pr{ν 1 = 0} = p independent of {X i, i 1}. Thus, N = d ν 1 + + ν l and by Lemma 3.1 S N = d S ν1 + + S νl, where S ν1,, S νl are independent random sums with S νk = d X 1 + + X νk, k = 1,, l. However, for any k = 1,, l, the probability mass function of S νk satisfies that for n = 1, 2,..., { νk } { νk } Pr {S νk = n} = (1 p) Pr X i = n ν k = 0 + p Pr X i = n ν k = 1 i=1 i=1 = p Pr{X 1 = n} and Pr {S νk = 0} = (1 p) + p Pr{X 1 = 0}. Thus, if Pr{X 1 = n} is decreasing in n = 1, 2,..., then Pr {S νk = n} is also decreasing in n = 1, 2,..., and hence D-PF 2 by Lemma 2.1 (i). Therefore, S N is D-PF 2 since D-PF 2 is preserved under convolution. This ends the proof of Theorem 3.1 Now, we prove that Theorem 3.1 holds for a discrete Poisson sum by approximating a Poisson random variable by a sequence of binomial random variables. Theorem 3.2 Let N be a Poisson random variable and {X i, i 1} be a sequence of i.i.d. counting random variables independent of N. If the probability mass function Pr{X 1 = n} is decreasing in n = 1, 2,..., then the Poisson sum S N = X 1 + + X N is D-PF 2. 7

Proof. Let Pr{N = k} = e λ λ k /k!, λ > 0, k = 0, 1,... be the probability mass function of N and {B n, n 1} be a sequence of binomial random variables independent of {X i, i 1} with the probability mass function ( ) ( ) k ( n λ Pr{B n = k} = 1 λ n k, k = 0, 1,..., n. (3.1) k n n) Without loss of generalization, we assume that n is sufficiently large so that 0 < λ/n < 1 in (3.1). Denote the probability generating functions (pgfs) of X 1, B n, and N by P (z), P n (z), and P N (z), respectively. Since the binomial sequence {B n } converge to the Poisson random variable N in distribution as n, we have for any z < 1, P n (z) P N (z), n. However, the pgfs of S Bn = X 1 + + X Bn and S N = X 1 + + X N are E(z S Bn ) = Pn (P (z)) and E(z S N ) = P N (P (z)), respectively. Hence, E(z S Bn ) E(z S N ), n, which implies that S Bn S N in distribution as n. Thus, if the probability mass function Pr{X 1 = n} is decreasing in n = 1, 2,..., then S Bn is D-PF 2 by Theorem 3.1. Hence, S N is D-PF 2 since D-PF 2 is preserved under weak convergence. Remark 3.1 We point out that both the DS-DFR and D-DFR properties are not preserved under Poisson sum. To see that, let N be a Poisson random variable with EN = λ > 0 and {X i, i 1} be a sequence of i.i.d. counting random variables independent of N. Further, let the pgf of X 1 be P (z) = E(z X 1 ) = n=0 p n z n, then the pgf of the Poisson random sum S N = X 1 + + X N is P λ (z) = e λ(p (z) 1). Since p 0 = Pr{X 1 = 0} = P (0) and p 1 = Pr{X 1 = 1} = P (0), we know that and Pr {S N = 0} = P λ (0) = e λ(p 0 1), Pr {S N = 1} = P λ(0) = λ p 1 e λ(p 0 1), Pr {S N = 2} = P λ (0) 2! ( (λp1 ) 2 ) = + λp 2 e λ(p0 1). 2 However, from Remark 2.2, we know that if the discrete distribution {p n, n = 0, 1, 2,...} is DS-DFR (D-DFR) then p n is decreasing in n = 0, 1,...(n = 1, 2,...). Thus, a Poisson sum S N = X 1 + + X N is not DS-DFR if Pr {S N = 0} < Pr {S N = 1}, which is equivalent to 1 < λp 1 or λ Pr{X 1 = 1} > 1, while a Poisson sum S N = X 1 + +X N is not D-DFR if Pr {S N = 1} < Pr {S N = 2}, which is equivalent to 2p 1 < λp 2 1 + 2p 2 or λ(pr{x 1 = 1}) 2 + 2 Pr{X 1 = 2} > 2 Pr{X 1 = 1}. (3.2) 8

Example 3.1 Let {X i, i = 1, 2, } be an i.i.d. counting random variables with Pr{X i = n} = 1 (n + 1)(n + 2), n = 0, 1, 2, and be independent of a Poisson random variable N with EN = λ > 6. Then X 1 D-LCVX DS-DFR D-DFR. However, (3.2) holds for any λ > 6 in this case. Hence, the Poisson sum X 1 + +X N is not D-DFR, and thus not D-LCVX or DS-DFR. This example shows that each of the classes of D-LCVX, DS-DFR, and D-DFR is not preserved under Poisson sum. However, this Poisson sum is D-PF 2 by Theorem 3.2 and hence it is an example of a counting random variable that is D-PF 2 but not D-DFR. Thus, D-DFR is a proper subclass of D-PF 2. Remark 3.2 We further remark that a discrete Poisson sum can yield both a D-LCVX random variable and a D-LCAV random variable. To see this fact, let {X i, i 1} be a sequence of i.i.d. logarithmic random variables with Pr{X 1 = n} = q n n log(1 q), n = 1, 2,, 0 < q < 1 and N be a Poisson random variable independent of {X i, i 1} with EN = α log(1 q), where α > 0. Then, the Poisson sum X 1 + + X N has a negative binomial distribution with ( ) k + α 1 Pr{X 1 + + X N = k} = (1 q) α q k, k = 0, 1,, k see, for example, Johnson et al (1992). However, the negative binomial distribution is D- LCVX if 0 < α 1 and D-LCAV if α 1. At the end of this section, we give the corresponding results for a continuous random sum X 1 + + X N, i.e. we assume that X 1 is supported on [0, ) in the rest of this section. We point out that the definition of PF 2 for a function defined on [0, ) is similar to that in Remark 2.1 for a function defined on {0, 1, 2,...}. See, for example, Karlin (1968) for details. Definition 3.1 A non-negative random variable X or its distribution F is said to be IFR, written X IFR, if F (x) is PF2 in x, or equivalently, if F (x + y)/ F (x) is decreasing in x 0 for any fixed y 0. On the other hand, a non-negative random variable X or its distribution F is said to be DFR, written X DFR if F (x + y)/ F (x) is increasing in x 0 for any fixed y 0. 9

Definition 3.2 A non-negative random variable X or its distribution F is said to be PF 2, written X PF 2, if F (x) is PF 2 in x, or equivalently, if F (x+y)/f (x) is decreasing in x 0 for any fixed y 0. Definition 3.3 A non-negative random variable X with a density function f(x) is said to have a log-convex density function, written X LCVX, if f(x) is log-convex, and is said to have a log-concave density function, written X LCAV, if f(x) is log-concave. Similarly to the results for D-PF 2 in Lemma 2.1, we can prove the corresponding results for PF 2. We just state the following Lemma 3.2 for PF 2 and omit their proofs since some of the proofs are similar to those for D-PF 2 and some of the results are well-known. Lemma 3.2 (i) For a distribution F supported on [0, ), if F has a decreasing density function on (0, ), then F is PF 2. (ii) DFR PF 2. (iii) LCAV PF 2. (vi) PF 2 is closed under convolution. (v) PF 2 is closed under weak convergence. We comment that for a continuous random sum S N = X 1 + + X N, if Pr{N = 0} > 0 then the random sum S N can not be IFR since in this case Pr{S N = 0} > 0, which means that the distribution function of S N is not continuous at the origin. However, a continuous random sum can be PF 2 even if Pr{N = 0} > 0. Theorem 3.3 Let N be a binomial random variable and {X i, i 1} be a sequence of i.i.d. non-negative random variables independent of N. If X 1 has a decreasing density function on (0, ), then the binomial sum S N = X 1 + + X N is PF 2. Proof. Similarly to the proof of Theorem 3.1, we denote the probability mass function of N by Pr{N = k} = ( ) l p k (1 p) l k, 0 < p < 1, k = 0, 1,, l. k Let ν 1,, ν l be i.i.d. Bernoulli random variables with Pr{ν 1 = 1} = 1 Pr{ν 1 = 0} = p independent of {X i, i 1}. Thus, N = d ν 1 + + ν l and by Lemma 3.1 S N = d S ν1 + + S νl, where S ν1,, S νl are independent random sums with S νk = d X 1 + + X νk, k = 1,, l. 10

However, for any k = 1,, l, the distribution function of S νk satisfies that for x 0, { νk } { νk } Pr {S νk x} = (1 p) Pr X i x ν k = 0 + p Pr X i x ν k = 1 i=1 = 1 p + p Pr{X 1 x}. Denote the distribution and density functions of X 1 by F (x) and f(x), respectively. Then, S νk is PF 2 if and only if for any 0 x 1 < x 2 and y 0, 1 p + pf (x 1 + y) 1 p + pf (x 1 ) holds. However, (3.3) is equivalent to 1 p + pf (x 2 + y) 1 p + pf (x 2 ) i=1 (1 p)[f (x 2 ) F (x 1 ) (F (x 2 + y) F (x 1 + y))] (3.3) + p [F (x 1 + y)f (x 2 ) F (x 1 )F (x 2 + y)] 0. (3.4) Thus, if f is decreasing on (0, ), then F is PF 2 by Lemma 3.2 (i) and hence F (x 1 + y)f (x 2 ) F (x 1 )F (x 2 + y) 0. Furthermore, F (x 2 ) F (x 1 ) (F (x 2 + y) F (x 1 + y)) = = x2 x 1 (f(t) f(t + y)) dt 0 if f is decreasing on (0, ). x2 x 1 f(t)dt x2 +y x 1 +y f(t)dt Therefore, if f is decreasing on (0, ), then (3.4) holds and hence S νk is PF 2. Thus, S N is PF 2 since PF 2 is preserved under convolution by Lemma 3.2(vi). This ends the proof of Theorem 3.3. Theorem 3.4 Let N be a Poisson random variable and {X i, i 1} be a sequence of i.i.d. non-negative random variables independent of N. If X 1 has a decreasing density function on (0, ), then the Poisson sum S N = X 1 + + X N is PF 2. Proof. The proof is similar to that of Theorem 3.2 by approximating the Poisson random variable by a sequence of binomial random variables and using Theorem 3.3 and Lemma 3.2(v). Remark 3.3 Using a different approach, Kijima and Nakagawa (1991) showed that a continuous Poisson sum X 1 + + X N is PF 2 if X 1 has a strictly decreasing density function f(x) on (0, ). The strictly decreasing condition is relaxed by the decreasing condition in Theorem 3.4. 11

4 The aging property of random sums In this section, we discuss the ageing property of a class of random sums. Let X be a counting random variable with probability distribution {p n = Pr{X = n}, n = 0, 1, 2,...} and mean 0 < EX <. The equilibrium distribution of the probability distribution {p n, n = 0, 1, 2,...} is defined by { p n = k=n+1 p k /EX = a n /EX, n = 0, 1,...}, where EX = n=0 np n = n=0 a n and a n = k=n+1 p k = Pr{X > n}. Furthermore, a counting random variable X is called an equilibrium random variable of X if the probability distribution of X is equal to the equilibrium distribution of X, namely Pr{ X = n} = p n = k=n+1 p k /EX, n = 0, 1,... Denote the pgf of X by P (z) = E(z X ) = p n z n, z < 1. n=0 It follows from Feller (1971) that the pgf P (z) of X satisfies P (z) = p n z n = n=0 1 P (z), z < 1. (4.1) (1 z)ex For a discrete random sum S N = X 1 + + X N. Denote the pgfs of N and X 1 by Q(z) and P (z), respectively. Then, the pgf of S N is Q(P (z)). Let S N be the equilibrium random variable of S N. Then (4.1) and ES N = EX 1 EN imply that the pgf of S N is 1 Q(P (z)) (1 z)es N = 1 Q(P (z)) (1 P (z))en 1 P (z). (4.2) (1 z)ex 1 Let Ñ be an equilibrium random variable of N independent of {X k, k = 1, 2,...} and denote the pgf of Ñ by Q(z). Thus, the pgf of the random sum X 1 + + XÑ is Q(P (z)) = where the equality follows from (4.1). 1 Q(P (z)) (1 P (z))en, Thus, (4.2) implies that the pgf of S N is the product of the pgfs of X 1 + + XÑ and X 1. Hence, S N = d SÑ + X 1, (4.3) where SÑ and X 1 are independent and SÑ = d X 1 + + XÑ. The decomposition (4.3) for an equilibrium random sum will be used to derive the NWUE aging property of a class of random sums. 12

Definition 4.1 A counting random variable X or its probability distribution {p n, n = 0, 1, 2,...} is said to be DS-NWU, written X DS-NWU, if a m+n+1 a m a n, n, m = 0, 1, 2,, (4.4) and is said to be DS-NWUE, written X DS-NWUE, if k=n+1 a k a n a k, n = 0, 1,..., k=0 which is equivalent to Pr{ X > n} Pr{X > n}, n = 0, 1,..., or equivalently, X st X, where X is an equilibrium random variable of X and a n = k=n+1 p k. It is clear that DS-NWU is a subclass of D-NWU since (4.4) implies a m+n+1 a n a m, n = 0, 1,. m=0 m=0 In addition, examples of DS-NWU can be found in Cai and Kalashnikov (2000). Furthermore, it is known that the class of DS-DFR is a subclass of DS-NWU. Thus, we have the following implications: DS-DFR DS-NWU DS-NWUE. The following theorem has been proved in Theorem 4.1 of Willmot and Cai (2001) for the DS-NWU aging property of a discrete random sum. Theorem 4.1 If N is DS-NWU, then the discrete random sum S N = X 1 + + X N is DS-NWU. Proof. See Theorem 4.1 of Willmot and Cai (2001). We now give the DS-NWUE aging property of a discrete random sum. Theorem 4.2 If N is DS-NWUE, then the discrete random sum S N = X 1 + + X N is DS-NWUE. 13

Proof. By (4.3), we have for any n = 0, 1,..., Pr{ S N > n} = Pr{SÑ + X 1 > n} Pr{SÑ > n} = Pr{X 1 + + XÑ > n}. (4.5) However, if N is DS-NWUE, then Ñ st N. Hence, X 1 + + XÑ st X 1 + + X N (4.6) since the stochastic order st is closed under random sum, see, for example, Shaked and Shanthikumar (1997). Therefore, (4.5) and (4.6) imply Pr{ S N > n} Pr{X 1 + + X N > n} = Pr{S N > n}, n = 0, 1,..., which implies that S N st S N. Thus, S N is DS-NWUE. Furthermore, the analogs of Theorems 4.1 and 4.2 for continuous random sums have been obtained in Cai and Kalashnikov (2002) and Willmot et al (2005), we restate them in the following Theorem 4.3 for completeness. Definition 4.2 A non-negative random variable X or its distribution F is said to be NWU if for any x 0 and y 0, F (x + y) F (x) F (y) and is said to be NWUE if 0 < EX < and for any x 0, x F (y)dy E(X) F (x). Theorem 4.3 If N is DS-NWU (DS-NWUE), then the continuous random sum S N = X 1 + + X N is NWU (NWUE). Proof. For the proof of the DS-NWU case see Theorem 3.1 of Cai and Kalashnikov (2000) and for the proof of the DS-NWUE case see Corollary 2.1 of Willmot et al (2005). 5 Concluding remarks We point out that any discrete equilibrium distribution has a decreasing probability mass function { k=n+1 p k / n=0 np n, n = 0, 1, 2,...} while any continuous equilibrium distribution has a decreasing density function F (x)/ 0 F (t)dt, x 0. Hence, there are many distributions satisfying the PF 2 conditions of decreasing probability mass functions or decreasing density functions in Theorems 3.1, 3.2, 3.3, and 3.4. Indeed, discrete and continuous equilibrium distributions are important probability distributions in statistics and probability and often appear in reliability, queueing, ruin theory, and other applied probability models. 14

Unlike a geometric sum, not much attention has been paid to the monotonicity and aging properties of a Poisson sum. There are still many unsolved problems for a Poisson sum. For example, we know that both DS-DFR and DFR properties are preserved under geometric summation, but we do not know any preservation results about a Poisson sum. On the other hand, it is interesting to notice that the aging properties of many random sums can be characterized uniquely by the aging properties of the primary distributions of the random sums whatever the underlying distributions of the random sums are. See, for example, the results given in Theorems 4.1, 4.2, and 4.3. References [1] Anderson, P.K., Borgan, O., Gill, R.D., and Keiding, N. (1993) Statistical models based on counting processes. Springer-Verlag, New York. [2] Brown, M. (1990) Error bounds for exponential approximations of geometric convolutions. The Annals of Probability 18, 1388-1402. [3] Cai, J. and Kalashnikov, V. (2000) NWU property of a class of random sums. Journal of Applied Probability 37, 283-289. [4] Feller, W. (1971) An Introduction to Probability Theory and Its Applications, Vol.II, 2nd Edition. Wiley, New York. [5] Johnson, N.L., Kotz, S. and Kemp, A.W. (1992) Univariate Discrete Distributions. 2nd edition, John Wiley & Sons, New York. [6] Kalbfleish, J.D. and Lawless, J.F. (1989) Inference based on retrospective ascertainment: An analysis of the data on transfusion-related AIDS. Journal of the American Statistical Association 84, 360-372. [7] Karlin, S. (1968) Total Positivity, Stanford University Press, Stanford, California. [8] Kijima, M. (1992) Further monotonicity properties of renewal processes. Advances in Applied Probability 25, 575-588. [9] Kijima, M. and Nakagawa, T. (1991) A cumulative damage shock model with imperfect preventive maintenance. Naval Research Logistic 38, 145-156. [10] Kováts, A. and Móri, T. (1992) Ageing properties of certain dependent geometric sums. Journal of Applied probability 29, 655-666. 15

[11] Lin, G.D. (2002) On the moment determinacy of the distributions of compound geometric sums. Journal of Applied probability 39, 545-554. [12] Nanda, A.K. and Shaked, M. (2001) The hazard rate and the reversed hazard rate orders, with applications to order statistics. Ann. Inst. Statist. Math. 53, 853-864. [13] Shaked, M. and Shanthikumar, J. (1994) Stochastic Orders and their Applications. Academic Press, San Diego. [14] Shanthikumar, J.G. (1988) DFR property of first passage times and its preservation under geometric compounding. The Annals of Probability 16, 397-406. [15] Steutel, F.W. and Van Harn, K. (2004) Infinite Divisibility of Probability Distributions on the Real Line, Marcel, Dekker, Inc., New York. [16] Willmot, G. (2002) On higher-order properties of compound geometric distributions. Journal of Applied Probability 39, 324-340. [17] Willmot, G. and Cai, J. (2001) Aging and other distributional properties of discrete compound geometric distributions. Insurance: Mathematics and Economics 28, 361-379. [18] Willmot, G. and Cai, J. (2004) On applications of residual lifetimes of compound geometric convolutions. Journal of Applied Probability 41, 802-815. [19] Willmot, G.E., Drekic, S., Cai, J. (2005) Equilibrium compound distributions and stoploss moments. Scandinavian Actuarial Journal, to appear. 16