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

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

2011-03-27

arxiv.org/abs/1103.5228v2

Mathematics

Probability

A newer version with added section about the periodic case

Scientific paper

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.

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