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

TL;DR

This paper establishes conditions under which the distribution of a random sum of independent, possibly non-identically distributed variables, with a random number of terms, exhibits an $ ext{O}$-exponential tail, broadening understanding of tail behaviors.

Contribution

It provides new sufficient conditions for the distribution of random sums to belong to the class of $ ext{O}$-exponential distributions, extending previous results to inhomogeneous sums.

Findings

Distribution function of the sum has $ ext{O}$-exponential tail under specified conditions.

Results apply to sums with non-identically distributed summands and random number of terms.

Broadens the class of distributions known to have $ ext{O}$-exponential tails.

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.