Limit theorem for reflected random walks

Théorème limite pour une marche aléatoire réfléchie

Hoang-Long Ngo (1) , Marc Peigné (2)

1 HNUE - Hanoi National University of Education
2 IDP - Institut Denis Poisson

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.

Mots clés: Invariance principle; Reflected Brownian motion; Renewal function

Domaines: Probabilités [math.PR]; Systèmes dynamiques [math.DS]

Hoang-Long Ngo, Marc Peigné. Limit theorem for reflected random walks. 2019. ⟨hal-02303955v1⟩