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

Let $S_n$ be a lattice random walk with mean zero and finite variance, and let ${\mathrm{\Lambda }}_n^a$ be its occupation measure at level $a$. In this note, we prove local limit theorems for $\mathrm{Pr}[S_n=x,{\mathrm{\Lambda }}_n^a=\mathrm{\ell }]$ and $\mathrm{Pr}[S_n=x\mathrm{\mid }{\mathrm{\Lambda }}_n^a=\mathrm{\ell }]$ in the cases where $a$, $\mathrm{\mid }x-a\mathrm{\mid }$ and $\mathrm{\ell }$ 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.