Random convolution of inhomogeneous distributions with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">O</mi></math>-exponential tail

Abstract

Let $\{\xi _{1},\xi _{2},\dots \}$ 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 $\{\xi _{1},\xi _{2},\dots \}$ and η 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.

Keywords

• Heavy tail
• exponential tail
• <italic>O</italic>-exponential tail
• random sum
• random convolution
• inhomogeneous distributions