Random convolution of inhomogeneous distributions with   $\mathcal{O}$-exponential tail
  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. 
Svetlana Danilenko, Simona Pa\v{s}kauskait\.e, Jonas \v{S}iaulys
math.PR