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

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.