The probability of reaching a receding boundary by branching random walk with fading branching and heavy-tailed jump distribution

TL;DR

This paper extends the analysis of receding boundary probabilities to branching random walks with fading branching and heavy-tailed jumps, providing tail asymptotics and uniform results over random time intervals.

Contribution

It introduces new tail asymptotics for the maximum of branching random walks with fading branching and heavy-tailed jumps, generalizing previous results to broader settings.

Findings

Derived tail asymptotics for the maximum of branching random walks with fading branching.

Established uniform results over bounded random time intervals.

Extended previous models to include heavy-tailed jump distributions and fading branching.

Abstract

Foss and Zachary (2003) and Foss, Palmowski and Zachary (2005) studied the probability of achieving a receding boundary on a time interval of random length by a random walk with a heavy-tailed jump distribution. They have proposed and developed a new approach that allows to generalise results of Asmussen (1998) onto the case of arbitrary stopping times and a wide class of nonlinear boundaries, and to obtain uniform results over all stopping times. In this paper, we consider a class of branching random walks with fading branching and obtain results on the tail asymptotics for the maximum of a branching random walk on a time interval of random (possibly unlimited) length, as well as uniform results within a class of bounded random time intervals.