Renenewal Approach to U-statistics Formarkovian Data

Consider a Markov chain X assumed to be positive recurrent with limiting probability distribution used to construct a U-statistics. Whereas the asymptotic properties of U -statistics based on independent and identically distributed data are well understood since the sixties the study of this speci…c class of statistics for dependent data has recently received special attention in the statistical literature.The purpose of this paper is to develop an alternative to the coupling methodology, speci…cally tailored for regenerative processes or stochastic processes for which a regenerative extension may be built, namely pseudo-regenerative processes. Indeed, sample paths of a Harris chain may be classically divided into i.i.d. regeneration blocks, namely data segments between random times at which the chain forgets its past. We develop further this view, in order to accurately investigate the asymptotic properties of U -statistics of positive Harris chains. A Strong Law of Large Numbers, Central Limit Theorem and Berry-Esseen bounds are established for markovian U -statistics under weak hypotheses. Following [4], we also propose to bootstrap certain markovian U statistics, using a speci…c resampling procedure, producing bootstrap data series with a renewal structure mimicking that of the original chain. 1 in ria -0 03 86 72 4, v er si on 1 22 M ay 2 00 9 Manuscrit auteur, publié dans "41èmes Journées de Statistique, SFdS, Bordeaux (2009)"