Return Probabilities for the Reflected Random Walk on $\mathbb N_0$

TL;DR

This paper analyzes the asymptotic probabilities of the reflected random walk on non-negative integers, revealing different decay rates depending on the drift of the underlying step distribution.

Contribution

It provides precise asymptotic formulas for the return probabilities of the reflected random walk under mild conditions, distinguishing cases based on the mean of the step distribution.

Findings

Exponential decay with polynomial correction when mean is positive.

Polynomial decay when mean is zero.

Explicit constants for asymptotic probabilities.

Abstract

Let $(Y_n)$ be a sequence of i.i.d. $\mathbb Z$-valued random variables with law $\mu$. The reflected random walk $(X_n)$ is defined recursively by $X_0=x \in \mathbb N_0, X_{n+1}=|X_n+Y_{n+1}|$. Under mild hypotheses on the law $\mu$, it is proved that, for any $ y \in \mathbb N_0$, as $n \to +\infty$, one gets $\mathbb P_x[X_n=y]\sim C_{x, y} R^{-n} n^{-3/2}$ when $\sum_{k\in \mathbb Z} k\mu(k) >0$ and $\mathbb P_x[X_n=y]\sim C_{y} n^{-1/2}$ when $\sum_{k\in \mathbb Z} k\mu(k) =0$, for some constants $R, C_{x, y}$ and $C_y >0$.