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Some aspects of fluctuations of random walks on R and applications to random walks on R+ with non-elastic reflection at 0

Abstract : In this article we refine well-known results concerning the fluctuations of one-dimensional random walks. More precisely, if $(S_n)_{n \geq 0}$ is a random walk starting from $0$ and $r\geq 0$, we obtain the precise asymptotic behavior as $n\to\infty$ of $\mathbb P[\tau^{>r}=n, S_n\in K]$ and $\mathbb P[\tau^{>r}>n, S_n\in K]$, where $\tau^{>r}$ is the first time that the random walk reaches the set $]r,\infty[$, and $K$ is a compact set. Our assumptions on the jumps of the random walks are optimal. Our results give an answer to a question of Lalley stated in \cite{L}, and are applied to obtain the asymptotic behavior of the return probabilities for random walks on $\mathbb R^+$ with non-elastic reflection at $0$.
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Submitted on : Friday, June 28, 2013 - 11:07:45 AM
Last modification on : Thursday, February 7, 2019 - 4:52:08 PM
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Rim Essifi, Marc Peigné, Kilian Raschel. Some aspects of fluctuations of random walks on R and applications to random walks on R+ with non-elastic reflection at 0. ALEA : Latin American Journal of Probability and Mathematical Statistics, Instituto Nacional de Matemática Pura e Aplicada, 2013, 10 (2), pp.591-607. ⟨hal-00780453v2⟩

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