Random convolution of inhomogeneous distributions with $\mathcal{O}$-exponential tail

Abstract: Let ${\xi_1,\xi_2,\ldots}$ be a sequence of independent random variables (not necessarily identically distributed), and $\eta$ be a counting random variable independent of this sequence. We obtain sufficient conditions on ${\xi_1,\xi_2,\ldots}$ and $\eta$ under which the distribution function of the random sum $S_{\eta}=\xi_1+\xi_2+\cdots+\xi_{\eta}$ belongs to the class of $\mathcal{O}$-exponential distributions.