Some aspects of fluctuations of random walks on R and applications to random walks on R+ with non-elastic reflection at 0
Résumé
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 $0$.
Origine : Fichiers produits par l'(les) auteur(s)
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