Asymptotics of convolution with the semi-regular-variation tail and its application to risk

TL;DR

This paper characterizes the tail behavior of a new subclass of exponential distributions related to semi-regular variation, proves closure under convolution, and applies these findings to estimate ruin probabilities in risk models.

Contribution

It introduces the semi-regular-variation-tailed distribution class, establishes its convolution properties, and applies the results to risk assessment without requiring identical distributions.

Findings

Derived precise tail asymptotics for the convolution of the new distribution class.

Proved the class is closed under convolution operations.

Provided asymptotic estimates for finite-time ruin probabilities in risk models.

Abstract

In this paper, according to a certain criterion, we divide the exponential distribution class into three subclasses. One of them is closely related to the regular-variation-tailed distribution class, so it is called the semi-regular-variation-tailed distribution class. In the class, although all distributions are not convolution equivalent,they still have some good properties. We give the precise tail asymptotic expression of convolution of these distributions, and prove that the new class is closed under convolution. In addition, we do not need to require the corresponding random variables to be identically distributed. Finally, we apply these results to a discrete time risk model with stochastic returns, and obtain the precise asymptotic estimation of the finite time ruin probability.