Martingale approach to subexponential asymptotics for random walks

TL;DR

This paper derives asymptotic probabilities for the supremum of a negative-mean random walk with long-tailed increments using martingale methods, avoiding prior distribution theory knowledge.

Contribution

It introduces a martingale-based approach to obtain subexponential asymptotics for the supremum of random walks, applicable to stopping times.

Findings

Asymptotics for P(M>x) as x→∞ derived

Asymptotics for P(M_τ>x) obtained

Method avoids reliance on long-tailed distribution theory

Abstract

Consider the random walk $S_n=\xi_1+...+\xi_n$ with independent and identically distributed increments and negative mean $\mathbf E\xi=-m<0$. Let $M=\sup_{0\le i} S_i$ be the supremum of the random walk. In this note we present derivation of asymptotics for $\mathbf P(M>x), x\to\infty$ for long-tailed distributions. This derivation is based on the martingale arguments and does not require any prior knowledge of the theory of long-tailed distributions. In addition the same approach allows to obtain asymptotics for $\mathbf P(M_\tau>x)$, where $M_\tau=\max_{0\le i<\tau}S_i$ and $\tau=\min\{n\ge 1: S_n\le 0 \}$.