Large Deviations for Systems of Noninteracting Recurrent Particles
Lee, Tzong-Yow
Ann. Probab., Tome 17 (1989) no. 4, p. 46-57 / Harvested from Project Euclid
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.
Publié le : 1989-01-14
Classification:  Large deviations,  empirical density,  infinite particle system,  recurrence,  60F10
@article{1176991493,
     author = {Lee, Tzong-Yow},
     title = {Large Deviations for Systems of Noninteracting Recurrent Particles},
     journal = {Ann. Probab.},
     volume = {17},
     number = {4},
     year = {1989},
     pages = { 46-57},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1176991493}
}
Lee, Tzong-Yow. Large Deviations for Systems of Noninteracting Recurrent Particles. Ann. Probab., Tome 17 (1989) no. 4, pp.  46-57. http://gdmltest.u-ga.fr/item/1176991493/