Local limit theorems for conditioned random walks by the heat kernel approximation

Abstract

We study the random walk $(S_n{)}_{n\ge 1}$ with independent and identically distributed real-valued increments having zero mean and an absolute moment of order $2+\hat{I}´$ for some $\hat{I}´>0$. For any starting point $x\in \mathbb{R}$, let $\ddot{I}{„}_x=\inf \{k\ge 1:x+S_k<0\}$ denote the first exit time of the random walk $x+S_n$ from the half-line $[0,\infty )$. In the previous work [25], we established a Gaussian heat kernel approximation for both the persistence probability $\mathbb{P}(\ddot{I}{„}_x>n)$ and the joint distribution $\mathbb{P}(x+S_n\le \cdot ,\ddot{I}{„}_x>n)$, uniformly over $x\in \mathbb{R}$ as $n\to \infty$. In this paper, we leverage these results to establish a novel conditioned local limit theorem for the walk $(x+S_n{)}_{n\ge 1}$. For $\mathbb{Z}$-valued random walks, we prove that the joint probability $\mathbb{P}(x+S_n=y,\ddot{I}{„}_x>n)$ is uniformly approximated by a distribution governed by the Gaussian heat kernel over all $x,y\in \mathbb{Z}$ as $n\to \infty$. Our new asymptotic unifies into a single comprehensive formula the classical local limit theorem by Caravenna [6], as well as various results relying on specific assumptions on $x$ and $y$. As a corollary, we obtain a new uniform-in-$x$ asymptotic formula for the local probability $\mathbb{P}(\ddot{I}{„}_x=n)$. We also extend our analysis to non-lattice random walks.

51 pages, 4 figures