Regeneration-based statistics for Harris recurrent Markov chains

Harris Markov chains make their appearance in many areas of statistical modeling, in particular in time series analysis. Recent years have seen a rapid growth of statistical techniques adapted to data exhibiting this particular pattern of dependence. In this paper an attempt is made to present how renewal properties of Harris recurrent Markov chains or of specific extensions of the latter may be practically used for statistical inference in various settings. When the study of probabilistic properties of general Harris Markov chains may be classically carried out by using the regenerative method (cf Smith (1955)), via the theoretical construction of regenerative extensions (see Nummelin (1978)), statistical methodologies may also be based on regeneration for general Harris chains. In the regenerative case, such procedures are implemented from data blocks corresponding to consecutive observed regeneration times for the chain. And the main idea for extending the application of these statistical techniques to general Harris chains X consists in generating first a sequence of approximate renewal times for a regenerative extension of X from data X1, ..., Xn and the parameters of a minorization condition satisfied by its transition probability kernel, and then applying the latter techniques to the data blocks determined by these pseudo-regeneration times as if they were exact regeneration blocks. Numerous applications of this estimation principle may be considered in both the stationary and nonstationary (including the null recurrent case) frameworks. This article deals with some important procedures based on (approximate) regeneration data blocks, from both practical and theoretical viewpoints, for the following topics: mean and variance estimation, confidence intervals, U -statistics, Bootstrap, robust estimation and statistical study of extreme values. 2 Patrice Bertail and Stéphan Clémençon