This article is cited in 3 papers

Short Communications

On summing a random number of random variables with increasing hazard rate or with strongly unimodal discrete distribution

O. P. Vinogradov

Moscow

Abstract: Let $\xi_1,\dots,\xi_n,\dots$ be independent identically distributed random variables and let $F(t)=\mathbf P\{\xi_i<t\}$ have an inscreasing hazard rate (IHR) [1]. The random sum $\zeta=\xi_1+\dots+\xi_{\tau}$ is considered where $\tau$ is independent of $\xi_i$ and the distribution of $\tau$ has also an IHR.
We find conditions under which the distribution of $\zeta$ has an IHR. The case of discrete $\xi_i$ is also considered. Analogous results for strongly unimodal discrete distributions are given.

Received: 13.06.1974

 English version: Theory of Probability and its Applications, 1976, 2:1, 205–209

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