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

By Michael Bromberg and Zemer Kosloff

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.Comment: This is the pre galley proof version of the article; Journal of Theoretical Probability 201

Topics: Mathematics - Probability, 60F17, 60J10
Year: 2012
OAI identifier: oai:arXiv.org:1103.5228
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