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

In this article we refine well-known results concerning the fluctuations of one-dimensional random walks. More precisely, if $(S_n{)}_{n\ge 0}$ is a random walk starting from 0 and $r\ge 0$, we obtain the precise asymptotic behavior as $n\to \mathrm{\infty }$ 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 $]r,\mathrm{\infty }[$, 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 [9], and are applied to obtain the asymptotic behavior of the return probabilities for random walks on ${\mathbb{R}}^{+}$ with non-elastic reflection at 0.