Weak invariance principle for the local times of partial sums of Markov Chains

TL;DR

This paper proves that the normalized local times of partial sums of a finite state Markov Chain converge in distribution to the local time of Brownian motion, under certain conditions.

Contribution

It establishes a weak invariance principle for the local times of Markov Chain partial sums, extending classical results to this setting.

Findings

Normalized local times converge to Brownian local time

Results hold under specific Markov Chain assumptions

Provides a functional limit theorem for local times

Abstract

Let X_{n} be an integer valued Markov Chain with finite state space. Let S_{n}=\sum_{k=0}^{n}X_{k} and let L_{n}(x) be the number of times S_{k} hits x up to step n. Define the normalized local time process t_{n}(x) by t_{n}(x)=\frac{L_{n}(\sqrt{n}(x)}{\sqrt{n}}. The subject of this paper is to prove a functional, weak invariance principle for the normalized sequence t_{n}, i.e. we prove that under some assumptions about the Markov Chain, the normalized local times converge in distribution to the local time of the Brownian Motion.