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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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