Conditioned local limit theorems for random walks on the real line

Abstract

Consider a random walk $S_n={\sum }_{i=1}^n{X}_i$ with independent and identically distributed real-valued increments $X_i$ of zero mean and finite variance. Assume that $X_i$ is non-lattice and has a moment of order $2+\delta$. For any $x\ge 0$, let ${\tau }_x=\inf \left\{k\ge 1:x+S_k<0\right\}$ be the first time when the random walk $x+S_n$ leaves the half-line $[0,\infty )$. We study the asymptotic behavior of the probability $\bb P (τ_x >n)$ and that of the expectation $\mathbb{E}\left(f(x+S_n),{\tau }_x>n\right)$ for a large class of target function $f$ and various values of $x$, $y$ possibly depending on $n$. This general setting implies limit theorems for the joint distribution $\mathbb{P}\left(x+S_n\in y+[0,\Delta ],{\tau }_x>n\right)$ where $\Delta >0$ may also depend on $n$. In particular, the case of moderate deviations $y=\sigma \sqrt{qn\log n}$ is considered. We also deduce some new asymptotics for random walks with drift and give explicit constants in the asymptotic of the probability $\bb P (τ_x =n)$. For the proofs we establish new conditioned integral limit theorems with precise error terms.

81 pages