Regenerative Block-bootstrap for Markov Chains

Patrice Bertail and Stéphan Clémençon
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Patrice Bertail: Crest
Stéphan Clémençon: Crest

No 2004-47, Working Papers from Center for Research in Economics and Statistics

Abstract: This paper introduces a specific Bootstrap method for positiverecurrent Markov chains, based on the regenerative method and the Nummelinsplitting technique. The main idea underlying this construction consists in generatinga sequence of approximate pseudo-renewal times for a Harris chain X fromdata X1; :::; Xn and the parameters of a minorization condition satisfied by itstransition probability kernel and then applying a variant of the methodology proposedby Datta &McCormick (1993) for bootstrapping additive functionals of typen¡1Pni=1 f(Xi)when the chain possesses an atom. We prove that, in the atomiccase, our method inherits the accuracy of the Bootstrap in the i.i.d. case up toOP(n¡1) under weak conditions. In the general (non necessarily) stationary case,asymptotic validity for this resampling procedure is established, provided that aconsistent estimator of the transition kernel may be computed. The second ordervalidity (up to a rate close to OP (n¡1)for regular stationary Markov Chains) isobtained in the stationary case. Applications to specific Markovian models arediscussed, together with some simulation results.

Date: 2004
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