Joint and Conditional Local Limit Theorems for Lattice Random Walks and Their Occupation Measures

TL;DR

This paper establishes local limit theorems for lattice random walks and their occupation measures, extending previous results and showing their asymptotic behavior matches that of Brownian motion and local time processes.

Contribution

It generalizes prior local limit theorems for lattice random walks by including occupation measures and their joint distributions with the walk's position.

Findings

Proves local limit theorems for joint probabilities involving the walk and occupation measures.

Shows asymptotic equivalence to Brownian motion and local time distributions.

Utilizes path decomposition methods to analyze random walks with independent increments.

Abstract

Let $S_n$ be a lattice random walk with mean zero and finite variance, and let $\Lambda^a_n$ be its occupation measure at level $a$. In this note, we prove local limit theorems for $\Pr[S_n=x,\Lambda^a_n=\ell]$ and $\Pr[S_n=x|\Lambda^a_n=\ell]$ in the cases where $a$, $|x-a|$ and $\ell$ are either zero or at least of order $\sqrt n$. The asymptotic description of these quantities matches the corresponding probabilities for Brownian motion and its local time process. This note can be seen as a generalization of previous results by Kaigh (1975) and Uchiyama (2011). In similar fashion to these results, our method of proof relies on path decompositions that reduce the problem at hand to the study of random walks with independent increments.