Regenerative Block-Bootstrap Confidence Intervals for the Extremal Index of Markov Chains

Abstract. This paper is devoted to show how the regenerative block-bootstrap methodology (RBB), proved asymptotically valid in the case of sample means and U-statistics based on data drawn from a regenerative Markov chain X = {Xn}n∈N in Bertail and Clémençon (2006a), Bertail and Clémençon (2006b), may be success-fully extended for constructing confidence intervals for the extremal index of in-stantaneous functions {f(Xn)}n∈N. Precisely, this boils down to applying the RBB procedure to the regenerative blocks estimator proposed in Bertail et al. (2007b) for measuring the clustering tendency of high threshold exceedances. Asymptotic normality of this estimator is established, together with the asymptotic validity of the bootstrap distribution estimate, under mild stochastic stability assumptions. Eventually, a preliminary numerical experiment is presented, with the aim to em-pirically evaluate the capacity of the RBB for building accurate confidence intervals for the extremal index of the waiting process related to a M/M/1 queue