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Limit theorem for reflected random walks

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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https://hal.archives-ouvertes.fr/hal-02303955
Contributor : Marc Peigné Connect in order to contact the contributor
Submitted on : Monday, February 8, 2021 - 11:55:59 AM
Last modification on : Tuesday, October 12, 2021 - 5:20:53 PM

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  • HAL Id : hal-02303955, version 3
  • ARXIV : 1910.01343

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Hoang-Long Ngo, Marc Peigné. Limit theorem for reflected random walks. 2019. ⟨hal-02303955v3⟩

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