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Large deviations and strong mixing. (English) Zbl 0863.60028

The authors prove a large deviation principle (LDP) with respect to the \(\tau\)-topology for the empirical measures of strongly mixing stationary processes taking values in a Polish space. The introduced strong mixing condition (S) is well chosen for the application of Hammersley’s approximate sub-additivity lemma. It is satisfied by any \(\alpha\)-mixing or \(\varphi\)-mixing stationary process with a hyper-exponential mixing rate of at least \(\alpha(n)\ll \exp(-n(\log n)^{1+\delta})\), for some \(\delta > 0\), or at least \(\varphi(n)\ll \exp(-nl(n))\) with \(l(n)\to \infty\) as \(n\to \infty\). The relation between (S) and hypermixing is discussed, and (S) is rephrased for Markov chains. Examples of positive recurrent Doeblin Markov chains, for which the LDP does not hold, indicate that the above conditions on the mixing rates are nearly optimal. They also illuminate the relations between the LDP for empirical means of all bounded \(\mathbb{R}^d\)-valued functionals and the LDP for the empirical measures.

MSC:

60F10 Large deviations
60G10 Stationary stochastic processes
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