# Maximum on a random time interval of a random walk with infinite mean

## TL;DR

This paper investigates the asymptotic behavior of the maximum of a random walk with infinite mean over a random time interval defined by the stopping time when the walk first drops below or equals zero.

## Contribution

It provides new asymptotic results for the probability that the maximum exceeds a large threshold in the context of random walks with infinite mean.

## Key findings

- Asymptotic characterization of P(M_τ > x) as x → ∞
- Results applicable to heavy-tailed distributions with infinite mean
- Insights into the maximum behavior of such random walks

## Abstract

Let $\xi_1,\xi_2,\ldots$ be independent, identically distributed random variables with infinite mean $\mathbf E[|\xi_1|]=\infty.$ Consider a random walk $S_n=\xi_1+\cdots+\xi_n$, a stopping time $\tau=\min\{n\ge 1: S_n\le 0\}$ and let $M_\tau=\max_{0\le i\le \tau} S_i$. We study the asymptotics for $\mathbf P(M_\tau>x),$ as $x\to\infty$.