Conditioned local limit theorems for random walks on the real line

TL;DR

This paper develops conditioned local limit theorems for zero-mean, finite variance, non-lattice random walks, analyzing their asymptotic behavior under various conditions and providing explicit constants and error estimates.

Contribution

It introduces new conditioned integral limit theorems with precise error terms for random walks, extending classical results to more general settings with explicit asymptotics.

Findings

Asymptotic behavior of $b P ( au_x >n)$ derived.

Limit theorems for joint distributions involving the walk and exit times established.

Explicit constants and error bounds provided for various asymptotic regimes.

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 $x\geq 0$, let $\tau_x = \inf \left\{ k\geq 1: x+S_{k} < 0 \right\}$ 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 $\bb P (\tau_x >n)$ 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 $y=\sigma \sqrt{q n\log n}$ is considered. We also deduce some new asymptotics for random walks with drift and give explicit constants in the asymptotic of the probability $\bb P (\tau_x =n)$. For the proofs we establish new conditioned integral limit theorems with precise error terms.