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Large deviations for systems of noninteracting recurrent particles. (English) Zbl 0664.60032

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 \(N\to \infty\), which a large class of transition probabilities does satisfy. Some features of our result are:
(a) The large deviation tails decay like \(\exp [-Nh^{-1}(N)I(\cdot)]\), more slowly than the known exp[-NI(\(\cdot)]\) 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.

MSC:

60F10 Large deviations
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