• Bernoulli
  • Volume 12, Number 4 (2006), 689-712.

Regenerative block bootstrap for Markov chains

Patrice Bertail and Stéphan Clémençon

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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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Bernoulli, Volume 12, Number 4 (2006), 689-712.

First available in Project Euclid: 16 August 2006

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bootstrap Edgeworth expansion Markov chain Nummelin splitting technique regenerative process


Bertail, Patrice; Clémençon, Stéphan. Regenerative block bootstrap for Markov chains. Bernoulli 12 (2006), no. 4, 689--712. doi:10.3150/bj/1155735932.

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  • [1] Asmussen, S. (1987) Applied Probability and Queues. Chichester: Wiley.
  • [2] Athreya, K. and Atuncar, G. (1998) Kernel estimation for real-valued Markov chains. Sankhya Ser. A, 60, 1-17.
  • [3] Athreya, K. and Fuh, C. (1989) Bootstrapping Markov chains: countable case. Technical Report B-89- 7, Institute of statistical Science, Academia Sinica, Taiwan.
  • [4] Bertail, P. and Clémençon, S. (2003) Regenerative block-bootstrap for Markov chains (Revised version). CREST preprint no. 2004-47. (accessed 10 February 2006).
  • [5] Bertail, P. and Clémençon, S. (2004) Edgeworth expansions for suitably normalized sample mean statistics of atomic Markov chains. Probab. Theory Related Fields, 130, 388-414.
  • [6] Bertail, P. and Clémençon, S. (2005) Note on the regeneration-based bootstrap for atomic Markov chains. Test. To appear.
  • [7] Bickel, P. and Freedman, D. (1981) Some asymptotic theory for the bootstrap. Ann. Statist., 9, 1196- 1217.
  • [8] Bolthausen, E. (1982) The Berry-Esseen theorem for strongly mixing Harris recurrent Markov chains. Z. Wahrscheinlichkeitstheorie Verw. Geb., 60, 283-289.
  • [9] Clémençon, S. (2000) Adaptive estimation of the transition density of a regular Markov chain. Math. Methods Statist., 9, 323-357.
  • [10] Clémençon, S. (2001) Moment and probability inequalities for sums of bounded additive functionals of regular Markov chains via the Nummelin splitting technique. Statist. Probab. Lett., 55, 227-238.
  • [11] Datta, S. and McCormick W. (1993) Regeneration-based bootstrap for Markov chains. Canad. J. Statist., 21, 181-193.
  • [12] Douc, R., Fort, G., Moulines E. and Soulier P. (2004) Practical drift conditions for subgeometric rates of convergence. Ann. Appl. Probab., 14, 1353-1377.
  • [13] Efron, B. (1979) Bootstrap methods: another look at the jackknife. Ann. Statist., 7, 1-26.
  • [14] Franke, J., Kreiss, J.P. and Mammen, E. (2002) Bootstrap of kernel smoothing in nonlinear time series. Bernoulli, 8, 1-37.
  • [15] Götze, F. and Künsch, H. (1996) Second order correctness of the blockwise bootstrap for stationary observations. Ann. Statist., 24, 1914-1933.
  • [16] Hall, P. (1992) The Bootstrap and Edgeworth Expansion. New York: Springer-Verlag.
  • [17] Hobert, J.P. and Robert, C.P. (2004) A mixture representation of with applications in Markov chain Monte Carlo and perfect sampling. Ann. Appl. Probab., 14, 1295-1305.
  • [18] Horowitz, J. (2003) Bootstrap methods for Markov processes. Econometrica, 71, 1049-1082.
  • [19] Jain, J. and Jamison, B. (1967) Contributions to Doeblin´s theory of Markov processes. Z. Wahrscheinlichkeitstheorie Verw. Geb., 8, 19-40.
  • [20] Kalashnikov, V. (1978) The Qualitative Analysis of the Behavior of Complex Systems by the Method of Test Functions. Moscow: Nauka.
  • [21] Lahiri, S. (2003) Resampling Methods for Dependent Data. New York: Springer-Verlag.
  • [22] Malinovskii, V. (1987) Limit theorems for Harris Markov chains I. Theory Probab. Appl., 31, 269-285.
  • [23] Malinovskii, V. (1989) Limit theorems for Harris Markov chains II. Theory Probab. Appl., 34, 252-265.
  • [24] Meyn, S. and Tweedie, R. (1996) Markov Chains and Stochastic Stability. London: Springer-Verlag.
  • [25] Nummelin, E. (1978) A splitting technique for Harris recurrent chains. Z. Wahrscheinlichkeitstheorie Verw. Geb., 43, 309-318.
  • [26] Orey, S (1971) Limit Theorems for Markov Chain Transition Probabilities. London: Van Nostrand Reinhold.
  • [27] Rachev, S. and Rüschendorf, L. (1998) Mass Transportation Problems. Vol. II: Applications. New York: Springer-Verlag.
  • [28] Roberts, G. and Rosenthal, J. (1996) Quantitative bounds for convergence rates of continuous time Markov processes. Electron. J. Probab., 9, 1-21.
  • [29] Smith, W. (1955) Regenerative stochastic processes. Proc. Roy. Soc., Lond. Ser. A, 232, 6-31.
  • [30] Thorisson, H. (2000) Coupling, Stationarity, and Regeneration. New York: Springer-Verlag.