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

2018/01/25 by Pierre Yves Gaudreau Lamarre, Lamarre, Pierre Yves Gaudreau
Decision Sciences · Mathematics · #60F05 #60G50 #60J55 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Probability and Risk Models #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1801.08469

openalex publication_date 2018/01/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let S_n be a lattice random walk with mean zero and finite variance, and let Λ^a_n be its occupation measure at level a. In this note, we prove local limit theorems for Pr[S_n=x,Λ^a_n=ℓ] and Pr[S_n=x|Λ^a_n=ℓ] in the cases where a, |x-a| and ℓ are either zero or at least of order √ 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.

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