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

Let $\{ξ_1,ξ_2,\ldots\}$ be a sequence of independent random variables (not necessarily identically distributed), and $η$ be a counting random variable independent of this sequence. We obtain sufficient conditions on $\{ξ_1,ξ_2,\ldots\}$ and $η$ under which the distribution function of the random sum $S_η=ξ_1+ξ_2+\cdots+ξ_η$ belongs to the class of $\mathcal{O}$-exponential distributions.

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