Mathematics – Probability
Scientific paper
2006-11-17
Annals of Probability 2007, Vol. 35, No. 4, 1307-1332
Mathematics
Probability
Published at http://dx.doi.org/10.1214/009171906000001024 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins
Scientific paper
10.1214/009171906000001024
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 $\mathbb{Z}^d$ tending to $\mathbb{Z}^d$ as time diverges. We finally discuss the relation of our density formula to the Ray--Knight theorem for continuous-time simple random walk on $\mathbb{Z}$, which is analogous to the well-known Ray--Knight description of Brownian local times.
Brydges David
der Hofstad Remco van
K{ö}nig Wolfgang
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