Asymptotics for the maximum of a modulated random walk with heavy-tailed increments

TL;DR

This paper investigates the asymptotic behavior of the maximum of a modulated random walk with heavy-tailed increments, especially when the modulating process is regenerative, and also examines related recursive maxima.

Contribution

It provides new asymptotic results for the maximum of modulated heavy-tailed random walks and recursive processes with regenerative modulating processes.

Findings

Derived asymptotic formulas for the maximum of the modulated random walk.

Analyzed the maximum of the recursive process with regenerative modulating process.

Extended existing theories to include heavy-tailed increments in modulated settings.

Abstract

We consider asymptotics for the maximum of a modulated random walk whose increments $\xi_n^{X_n}$ are heavy-tailed. Of particular interest is the case where the modulating process $X$ is regenerative. Here we study also the maximum of the recursion given by $W_0=0$ and, for $n\ge1$, $W_n=\max(0,W_{n-1}+\xi_n^{X_n})$