Limit theorem for reflected random walks

TL;DR

This paper proves that properly rescaled reflected random walks with certain finite moments converge in distribution to reflected Brownian motion, extending understanding of their long-term behavior.

Contribution

It establishes a limit theorem showing convergence of reflected random walks to reflected Brownian motion under specific moment and aperiodicity conditions.

Findings

Reflected walk converges to reflected Brownian motion under finite second moment.

The convergence holds when the increments are aperiodic and centered.

The result extends classical null-recurrence analysis to a functional limit theorem.

Abstract

Let $\xi$ n , n $\in$ N be a sequence of i.i.d. random variables with values in Z. The associated random walk on Z is S(n) = $\xi$ 1 + $\times$ $\times$ $\times$ + $\xi$ n+1 and the corresponding "reflected walk" on N 0 is the Markov chain X(n), n $\in$ N, given by X(0) = x $\in$ N 0 and X(n + 1) = |X(n) + $\xi$ n+1 | for n $\ge$ 0. It is well know that the reflected walk (X(n)) n$\ge$0 is null-recurrent when the $\xi$ n are square integrable and centered. In this paper, we prove that the process (X(n)) n$\ge$0 , properly rescaled, converges in distribution towards the reflected Brownian motion on R + , when E[$\xi$ 2 n ] < +$\infty$, E[(max(0, --$\xi$ n) 3 ] < +$\infty$ and the $\xi$ n are aperiodic and centered.