Large Deviations for Products of Empirical Measures of Dependent Sequences

#### P. Eichelsbacher, U. Schmock

2001, v.7, №3, 435-468

ABSTRACT

We prove large deviation principles (LDP) for $m$-fold products of empirical measures and for $\Ustat$-empirical measures, where the underlying sequence of random variables is a special Markov chain, an exchangeable sequence, a mixing sequence or an independent, but not identically distributed sequence. The LDP can be formulated on a subset of all probability measures, endowed with a topology which is even finer than the usual $\tau$-topology. The advantage of this topology is that the map $\nu\mapsto\int_{S^m}\varphi\,d\nu$ is continuous even for certain unbounded $\varphi$ taking values in a Banach space. As a particular application we get large deviation results for $\Ustat$-statistics and $\vstat$-statistics based on dependent sequences. Furthermore, we prove an LDP for products of empirical processes in a topology, which is finer than the projective limit $\tau$-topology.

Keywords: large deviations,exponential approximations,contraction principle,weak topology,$\tau$-topology,Markov chains,exchangeable sequences,mixing sequences,$\Ustat$-empirical measures,$\Ustat$-statistics,$\vstat$-statistics