Exit times for integrated random walks

TL;DR

This paper analyzes the asymptotic probability that the area under a centered random walk with finite variance stays positive up to large time n, revealing it decays as n^{-1/4} using discrete potential theory.

Contribution

It introduces a discrete potential theory approach to derive the exact asymptotics of the positivity probability for integrated random walks.

Findings

Probability decays as n^{-1/4} for large n

Finite (2+δ)-th moment assumption is sufficient

Develops a new discrete potential theory framework

Abstract

We consider a centered random walk with finite variance and investigate the asymptotic behaviour of the probability that the area under this walk remains positive up to a large time $n$. Assuming that the moment of order $2+\delta$ is finite, we show that the exact asymptotics for this probability are $n^{-1/4}$. To show these asymptotics we develop a discrete potential theory for the integrated random walk.