# On the long tail property of product convolution

## TL;DR

This paper investigates the conditions under which the product convolution of two independent non-negative random variables results in a long-tailed distribution, expanding understanding beyond exponential cases.

## Contribution

It establishes new sufficient conditions for the product convolution to be long-tailed, including cases where one distribution is generalized long-tailed, not necessarily exponential.

## Key findings

- Conditions for long-tailedness of product convolution are identified.
- Many distributions satisfy these conditions, with some shown to be necessary.
- Examples illustrate the applicability of the theoretical results.

## Abstract

Let $X$ and $Y$ be two independent random variables with corresponding distributions $F$ and $G$ supported on $[0,\infty)$. The distribution of the product $XY$, which is called the product convolution of $F$ and $G$, is denoted by $H$. In this paper, some suitable conditions about $F $ and $G $ are given, under which the distribution $H$ belongs to the long-tailed distribution class. Here, $F$ is a generalized long-tailed distribution and is not necessarily an exponential distribution. Finally, a series of examples are given to show that the above conditions are satisfied by many distributions and one of them is necessary in some sense.