Open Access
aug 2006 Regenerative block bootstrap for Markov chains
Patrice Bertail, Stéphan Clémençon
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Bernoulli 12(4): 689-712 (aug 2006). DOI: 10.3150/bj/1155735932


A specific bootstrap method is introduced for positive recurrent Markov chains, based on the regenerative method and the Nummelin splitting technique. This construction involves generating a sequence of approximate pseudo-renewal times for a Harris chain X from data X1,..., Xn and the parameters of a minorization condition satisfied by its transition probability kernel and then applying a variant of the methodology proposed by Datta and McCormick for bootstrapping additive functionals of type n-1i=1nf(Xi) when the chain possesses an atom. This novel methodology mainly consists in dividing the sample path of the chain into data blocks corresponding to the successive visits to the atom and resampling the blocks until the (random) length of the reconstructed trajectory is at least n, so as to mimic the renewal structure of the chain. In the atomic case we prove that our method inherits the accuracy of the bootstrap in the independent and identically distributed case up to OP(n-1) under weak conditions. In the general (not necessarily stationary) case asymptotic validity for this resampling procedure is established, provided that a consistent estimator of the transition kernel may be computed. The second-order validity is obtained in the stationary case (up to a rate close to OP(n-1) for regular stationary chains). A data-driven method for choosing the parameters of the minorization condition is proposed and applications to specific Markovian models are discussed.


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Patrice Bertail. Stéphan Clémençon. "Regenerative block bootstrap for Markov chains." Bernoulli 12 (4) 689 - 712, aug 2006.


Published: aug 2006
First available in Project Euclid: 16 August 2006

zbMATH: 1125.62037
MathSciNet: MR2248233
Digital Object Identifier: 10.3150/bj/1155735932

Keywords: bootstrap , Edgeworth expansion , Markov chain , Nummelin splitting technique , Regenerative process

Rights: Copyright © 2006 Bernoulli Society for Mathematical Statistics and Probability

Vol.12 • No. 4 • aug 2006
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