The local asymptotic estimation for the supremum of a random walk with generalized strong subexponential summands

TL;DR

This paper develops local asymptotic estimates for the maximum of a random walk with summands from new distribution classes that are long-tailed and generalized strong subexponential, expanding understanding of their tail behaviors.

Contribution

It introduces new distribution classes with favorable properties and analyzes the local asymptotic behavior of the supremum of a random walk with such summands.

Findings

New distribution classes with long-tailed and generalized strong subexponential properties

Identification of distributions that intersect with existing classes, expanding the scope of tail analysis

Establishment of local asymptotic estimates for the supremum of the random walk

Abstract

In this paper, the local asymptotic estimation for the supremum of a random walk and its applications are presented. The summands of the random walk have common long-tailed and generalized strong subexponential distribution. This distribution class and the corresponding generalized local subexponential distribution class are two new distribution classes with some good properties. Further, some long-tailed distributions with intuitive and concrete forms are found, which show that the intersection of the two above-mentioned distribution classes with long-tailed distribution class properly contain the strong subexponential distribution class and the locally subexponential distribution class, respectively.