Limit theorem for reflected random walks

2019/10/03 by Ngo, Hoang-Long, Peigné, Marc · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1910.01343

Abstract

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

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