Geodesic
On summing a~random number of random variables with increasing hazard rate or with strongly unimodal discrete distribution
Teoriâ veroâtnostej i ee primeneniâ, Tome 21 (1976) no. 1, pp. 209-214

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Let $\xi_1,\dots,\xi_n,\dots$ be independent identically distributed random variables and let $F(t)=\mathbf P\{\xi_i$ 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. 
@article{TVP_1976_21_1_a24,
     author = {O. P. Vinogradov},
     title = {On summing a~random number of random variables with increasing hazard rate or with strongly unimodal discrete distribution},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {209--214},
     publisher = {mathdoc},
     volume = {21},
     number = {1},
     year = {1976},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1976_21_1_a24/}
}
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O. P. Vinogradov. On summing a~random number of random variables with increasing hazard rate or with strongly unimodal discrete distribution. Teoriâ veroâtnostej i ee primeneniâ, Tome 21 (1976) no. 1, pp. 209-214. http://geodesic.mathdoc.fr/item/TVP_1976_21_1_a24/