Limit theorem for reflected random walks

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.