Some aspects of fluctuations of random walks on R and applications to random walks on R+ with non-elastic reflection at 0

TL;DR

This paper refines the understanding of fluctuations in one-dimensional random walks, providing precise asymptotics for certain hitting probabilities and applying these results to analyze return probabilities of reflected walks on the positive real line.

Contribution

It offers the exact asymptotic behavior of hitting probabilities for one-dimensional random walks with optimal assumptions, answering a question posed by Lalley and extending to reflected walks.

Findings

Asymptotic behavior of hitting probabilities as n→∞

Optimal assumptions on jump distributions

Application to return probabilities of reflected walks

Abstract

In this article we refine well-known results concerning the fluctuations of one-dimensional random walks. More precisely, if $(S_n)_{n \geq 0}$ is a random walk starting from 0 and $r\geq 0$, we obtain the precise asymptotic behavior as $n\to\infty$ of $\mathbb P[\tau^{>r}=n, S_n\in K]$ and $\mathbb P[\tau^{>r}>n, S_n\in K]$, where $\tau^{>r}$ is the first time that the random walk reaches the set $]r,\infty[$, and $K$ is a compact set. Our assumptions on the jumps of the random walks are optimal. Our results give an answer to a question of Lalley stated in [9], and are applied to obtain the asymptotic behavior of the return probabilities for random walks on $\mathbb R^+$ with non-elastic reflection at 0.