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July 2007 Joint density for the local times of continuous-time Markov chains
David Brydges, Remco van der Hofstad, Wolfgang König
Ann. Probab. 35(4): 1307-1332 (July 2007). DOI: 10.1214/009171906000001024

Abstract

We investigate the local times of a continuous-time Markov chain on an arbitrary discrete state space. For fixed finite range of the Markov chain, we derive an explicit formula for the joint density of all local times on the range, at any fixed time. We use standard tools from the theory of stochastic processes and finite-dimensional complex calculus.

We apply this formula in the following directions: (1) we derive large deviation upper estimates for the normalized local times beyond the exponential scale, (2) we derive the upper bound in Varadhan’s lemma for any measurable functional of the local times, and (3) we derive large deviation upper bounds for continuous-time simple random walk on large subboxes of ℤd tending to ℤd as time diverges. We finally discuss the relation of our density formula to the Ray–Knight theorem for continuous-time simple random walk on ℤ, which is analogous to the well-known Ray–Knight description of Brownian local times.

Citation

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David Brydges. Remco van der Hofstad. Wolfgang König. "Joint density for the local times of continuous-time Markov chains." Ann. Probab. 35 (4) 1307 - 1332, July 2007. https://doi.org/10.1214/009171906000001024

Information

Published: July 2007
First available in Project Euclid: 8 June 2007

zbMATH: 1127.60076
MathSciNet: MR2330973
Digital Object Identifier: 10.1214/009171906000001024

Subjects:
Primary: 60J27 , 60J55

Keywords: Density , large deviations upper bound , Local times , Ray–Knight theorem , Varadhan’s lemma

Rights: Copyright © 2007 Institute of Mathematical Statistics

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Vol.35 • No. 4 • July 2007
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