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

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