M.: 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 (Sn)n>0 is a random walk starting from 0 and r> 0, we obtain the precise asymptotic behavior as n→ ∞ of P[τ>r = n, Sn ∈ K] and P[τ>r> n, Sn ∈ K], where τ>r is the first time that the random walk reaches the set]r,∞[, 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 [12], and are applied to obtain the asymptotic behavior of the return probabilities for random walks on R+ with non-elastic reflection at 0. 1