## The Annals of Probability

- Ann. Probab.
- Volume 17, Number 1 (1989), 46-57.

### Large Deviations for Systems of Noninteracting Recurrent Particles

#### Abstract

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.

#### Article information

**Source**

Ann. Probab., Volume 17, Number 1 (1989), 46-57.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176991493

**Digital Object Identifier**

doi:10.1214/aop/1176991493

**Mathematical Reviews number (MathSciNet)**

MR972769

**Zentralblatt MATH identifier**

0664.60032

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F10: Large deviations

**Keywords**

Large deviations empirical density infinite particle system recurrence

#### Citation

Lee, Tzong-Yow. Large Deviations for Systems of Noninteracting Recurrent Particles. Ann. Probab. 17 (1989), no. 1, 46--57. doi:10.1214/aop/1176991493. https://projecteuclid.org/euclid.aop/1176991493