Precise local large deviations for random sums with applications

TL;DR

This paper derives precise local large deviation probabilities for sums of independent random variables with regularly varying distributions, with applications to risk models and claim surplus processes.

Contribution

It provides new results on local large deviations for random sums with regularly varying distributions and applies these to generalized risk models.

Findings

Derived explicit formulas for local large deviations.

Extended results to claim surplus processes in risk models.

Applicable to both finite and infinite interval cases.

Abstract

In this paper, we investigate the precise local large deviation probabilities for random sums of independent real-valued random variables with a common distribution $F$, where $F(x+\Delta)=F((x, x+T])$ is an $\mathcal{O}$-regularly varying function for some fixed constant $T>0$(finite or infinite). We also obtain some results on precise local large deviation probabilities for the claim surplus process of generalized risk models in which the premium income until time $t$ is simply assumed to be a nondecreasing and nonnegative stochastic process. In particular, the results we obtained are also valid for the global case, i.e. case $T=\infty$.