Open Access
January, 1989 Large Deviations for Systems of Noninteracting Recurrent Particles
Tzong-Yow Lee
Ann. Probab. 17(1): 46-57 (January, 1989). DOI: 10.1214/aop/1176991493


We consider noninteracting systems of infinite particles each of which follows an irreducible, null recurrent Markov process and prove a large deviation principle for the empirical density. The expected occupation time (up to time $N$) of this Markov process, named as $h(N)$, plays an essential role in our result. We impose on $h(N)$ a regularly varying property as $N \rightarrow \infty$, which a large class of transition probabilities does satisfy. Some features of our result are: (a) The large deviation tails decay like $\exp\lbrack - Nh^{-1}(N)I(\cdot)\rbrack$, more slowly than the known $\exp\lbrack - NI(\cdot) \rbrack$ type of decay in transient situations. (b) Our rate function $I(\lambda(\cdot))$ equals infinity unless $\lambda(\cdot)$ is an invariant distribution. (c) Our rate function is explicit and is rather insensitive to the underlying Markov process. For instance, if we randomized the time steps of a Markov chain by exponential waiting time of mean 1, the resultant system obeys exactly the same large deviation principle.


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Tzong-Yow Lee. "Large Deviations for Systems of Noninteracting Recurrent Particles." Ann. Probab. 17 (1) 46 - 57, January, 1989.


Published: January, 1989
First available in Project Euclid: 19 April 2007

zbMATH: 0664.60032
MathSciNet: MR972769
Digital Object Identifier: 10.1214/aop/1176991493

Primary: 60F10

Keywords: empirical density , Infinite particle system , large deviations , recurrence

Rights: Copyright © 1989 Institute of Mathematical Statistics

Vol.17 • No. 1 • January, 1989
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