The asymptotic hazard rate of sums of discrete random variables

Abstract

We show that the asymptotic hazard rate of the sum of discrete random variables is dominated by the smallest asymptotic failure rate of the summands.

Full text

The asymptotic hazard rate of sums of discrete random variables
Joachim Arts, Geert-Jan van Houtum, Bert Zwart∗
January 30, 2017
Abstract
We show that the asymptotic hazard rate of the sum of discrete random variables is dominated by
the smallest asymptotic failure rate of the summands.
AMS subject classification: 60E05.
1. Introduction and main result
The purpose of this note is to determine the asymptotic hazard rate of the sum of independent non-negative
discrete random variables. More precisely, we prove the following.
Proposition 1. Let X, Y be independent non-negative integer-valued random variables such that
P(X=n)/P (X≥n)→rand P(Y=n)/P (Y≥n)→sas n→ ∞,r, s ∈[0,1]. As n→ ∞,
P(X+Y=n)/P (X+Y≥n)→min{r, s}.
Proposition 1 is related to a classical convolution property of subexponential and related distributions;
in particular Theorem 3 of [4]; more recent formulations can be found in [2, 3]. In this article, we focus on
integer-valued distributions. Our motivation comes from an inventory model with lost sales in which ordered
inventory arrives after a large lead time n[1]. In that problem an approximation is developed that becomes
accurate when nis large, and the quality of the approximation turns out to depend on the result of the
proposition presented here.
2. Proof
Without loss of generality, r≤s. The proof will distinguish three cases: r= 1, r < s ≤1 and r=s < 1.
The case r < s ≤1:observe that the assumptions on Xand Yare equivalent to
P(X > n +m)
P(X > n)→(1 −r)m,P(Y > n +m)
P(Y > n)→(1 −s)m,
as n→ ∞ along the integers, for every integer m. This should be compared with the class of distributions
called L(γ), for γ≥0. A random variable Uis a member of L(γ) if, for all y > 0
P(U > x +y)
P(U > x)→e−γy ,(1)
∗JA and GJvH are at Eindhoven University of Technology (TU/e), department of Industrial Engineering. BZ (corresponding
author, bertz@cwi.nl), is at CWI Amsterdam and the department of Mathematics at TU/e. This research is partly funded by
NWO
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as x→ ∞ for all real y. Theorem 3 of [4] essentially states, that if U∈ L(γ) and V∈ L(δ) then U+V∈
L(min{γ, δ }).
Since our setting is restricted to integer-valued r.v.’s Xis not member of the class L(−ln(1 −r)) a formal
proof is needed. In addition, the inequality on the last line of page 253 of [4] is not correct. In what follows
we correct (and even simplify) that proof and have it apply to our discrete setting.
Set F(n) = P(X≤n), G(n) = P(Y≤n), H(n) = P(X+Y≤n), ¯
F(n) = 1 −F(n). ¯
Gand ¯
Hare
defined similarly. Observe now that, as in p. 252 of [4]:
0≤P(X+Y > n;Y > n −m)
P(X+Y > n;Y≤n−m)≤P(Y > n −m)
P(X > n;Y≤n−m)
=P(Y > n −m)
P(X > n −m)
P(X > n −m)
P(X > n)
P(X > n)
P(X > n)P(Y≤n−m)→0×(1 −r)−m×1=0.
Let f(n)∼g(n) denote f(n)/g(n)→1. The above implies that
P(X+Y > n)∼
n−m
X
k=0
P(X > n −k)P(Y=k),(2)
so that also (replacing mby m−land nby n−l)
P(X+Y > n −l)∼
n−m
X
k=0
P(X > n −k−l)
P(X > n −k)P(X > n −k)P(Y=k).(3)
Set now UX(j, l) = sup{n:P(X > n −l)/P (X > n); n≥j}and
LX(j, l) = inf{n:P(X > n −l)/P (X > n); n≥j}. The r.h.s. of (3) is between
LX(m, l)
n−m
X
k=0
P(X > n −k)P(Y=k) and UX(m, l)
n−m
X
k=0
P(X > n −k)P(Y=k).
Since limm→∞ LX(m, l) = limm→∞ UX(m, l) = (1 −r)−l, we obtain from (3),
P(X+Y > n −l)∼(1 −r)−l
n−m
X
k=0
P(X > n −k)P(Y=k)∼(1 −r)−lP(X+Y > n).(4)
The last equivalence follows from (2).
The case r=s < 1:define LYand UYsimilarly as LXand UX. As (2.12) of [4] is also valid for integer-valued
r.v. we simply state its conclusion. For any m > 0:
P(X+Y > n −l)≤max{UX(m, l), UY(n−m+l, l)}P(X+Y > n).(5)
Thus, we see that
lim sup
n→∞
P(X+Y > n −l)
P(X+Y > n)≤max{UX(m, l),(1 −r)−l},
for every m > 0. Now let m→ ∞ to conclude that
lim sup
n→∞
P(X+Y > n −l)
P(X+Y > n)≤(1 −r)−l.
The proof of the lower bound is similar.
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The case r= 1:let ∈(0,1) and Kbe such that P(X=i)≥(1 −)P(X≥i) for i≥K. Write for
n > 2K:
P(X+Y=n)≥
n
X
i=K
P(X=i)P(Y=n−i)≥(1 −)
n
X
i=K
P(X≥i)P(Y=n−i)
≥(1 −)P(X+Y≥n;Y≤n−K) = (1 −)[P(X+Y≥n)−P(X+Y≥n;Y > n −K)]
≥(1 −)[P(X+Y≥n)−P(Y > n −K)].
Thus,
1≥P(X+Y=n)
P(X+Y≥n)≥(1 −)1−P(Y > n −K)
P(X+Y≥n).
Since P(Y > n −K)
P(X+Y≥n)≤P(Y > n −K)
P(Y > n −K−1)P(X < K + 1) →0
given the assumption on Y, we conclude that
lim inf
n→∞
P(X+Y=n)
P(X+Y≥n)≥1−.
The proof is now complete by letting ↓0.
Acknowledgment
The authors are grateful to a referee for a careful reading of the manuscript. The research of the last author
is supported by VICI grant 639.033.413 of NWO.
References
[1] Arts, J., Houtum, G.J., Levi, R. (2017). Base-stock Policies for Lost Sales Models: Aggregation and
Asymptotics. In preparation.
[2] Block, H. W., Langberg, N. A., Savits, T. H. (2014). The limiting failure rate for a convolution of gamma
distributions. Statistics and Probability Letters 94, 176–180.
[3] Block, H. W., Langberg, N. A. and Savits, T. H. (2015). The limiting failure rate for a convolution of
life distributions. Journal of Applied Probability 52, 894–898.
[4] Embrechts, P., Goldie, C. (1980). On closure and factorization properties of subexponential and related
distributions. J. Austral. Math. Soc. (A) 29, 243–256.
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