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

Abstract

Let $\left( X_{i}\right) _{i\ge 1}$ be a sequence i.i.d.  random variables and $S_{n}=\sum _{i=1}^{n}X_{i}$, $n\ge 1$. For any starting point $y>0$ denote by $\tau _{y}$ the first moment when the random walk $\left( y+S_{k}\right) _{k\ge 1}$ becomes negative. We give some bounds of order $n^{-3/2}$ for the expectations $\mathbb {E}\left( g\left( y+S_{n}\right) ;\tau _{n}>n\right)$, $y\in \mathbb {R}_{+}^{*}$ which are valid for a large class of bounded measurable function g with constants depending on some norms of the function g.