Large Deviations for Products of Empirical Measures of Dependent Sequences

Peter Eichelsbacher and Uwe Schmock

Abstract: We prove large deviation principles (LDP) for m-fold products of empirical measures and for U-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 taking the integral of f is a continuous operation even for certain unbounded f taking values in a Banach space. As a particular application we get large deviation results for U-statistics and V-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, U-empirical measures, U-statistics, V-statistics

Reference: Markov Processes and Related Fields, Vol. 7, No. 3 (2001) 435-468.

DOI: 10.1023/A:1022610532538

2010 Mathematics Subject Classification:

The paper (26 pages, revised version May 12, 2001) is available in:

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