Bounds in the Local Limit Theorem for a Random Walk Conditioned to Stay Positive

Abstract

Let (Xi)i≥1 be a sequence i.i.d. random variables and Sn=∑i=1nXi, n≥ 1. For any starting point y> 0 denote by τy the first moment when the random walk (y+Sk)k≥1 becomes negative. We give some bounds of order n- 3 / 2 for the expectations E(g(y+ Sn) ; τn> n), y∈R+∗ which are valid for a large class of bounded measurable function g with constants depending on some norms of the function g.