Regenerative bootstrap for U-statistics and extremes.

Abstract

The purpose of this talk is to present an alternative to the coupling methodology, specifically tailored for regenerative processes or stochastic processes for which a regenerative extension may be built, namely pseudo-regenerative processes. This includes the case of general Harris Markov chains on which the present study focuses. 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, termed regeneration times. Hence, many results established in the i.i.d. setup may be extended to the markovian framework by applying the latter to (functionals of) the regeneration blocks. Refer to MeynTweedie's Book, for the Strong Law of Large Numbers and the Central Limit Theorem, as well as Bolthausen, 80, Malinovskii, 87, Malinovskii, 89, Bertail and Clémençon 2004-2008 for refinements of the CLT. This approach to the study of the behavior of Markov chains based on renewal theory is known as the regenerative method. In the present article, we develop further this view, in order to accurately investigate the asymptotic properties of U-statistics of positive Harris chains. Our approach crucially relies on the notion of regenerative U-statistic approximate of a markovian U-statistic. As the approximant is a standard U-statistic based on regeneration blocks, classical theorems apply to the latter and consequently yield the corresponding results for the original statistic. Beyond gaussian asymptotic confidence intervals, we propose to bootstrap certain markovian U statistics, using the specific resampling procedure introduced in Bertail and Clémençon 2005, 2006 producing bootstrap data series with a renewal structure mimicking that of the original chain. The asymptotic validity of the (approximate) regenerative block-bootstrap of U-statistics is rigorously established. For illustration purpose, some simulation results are displayed. We will also show rapidly how the same kind of techniques allows to obtain consistent estimator of the tail index and of the dependence index.