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Journal Articles Lecture Notes in Mathematics Year : 2021

Limit theorem for reflected random walks

Théorème limite pour une marche aléatoire réfléchie

Hoang-Long Ngo
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Marc Peigné

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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Dates and versions

hal-02303955 , version 1 (02-10-2019)
hal-02303955 , version 2 (05-02-2021)
hal-02303955 , version 3 (08-02-2021)

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Hoang-Long Ngo, Marc Peigné. Limit theorem for reflected random walks. Lecture Notes in Mathematics, 2021, ⟨10.1007/978-3-030-74863-0_6⟩. ⟨hal-02303955v3⟩
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