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 (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 (1995), and are applied to obtain the asymptotic behavior of the return probabilities for random walks on R + with non-elastic reflection at 0.