## Regenerative block-bootstrap for Markov chains.(English)Zbl 1125.62037

Summary: 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 $$X_1,\dots,X_n$$ and the parameters of a minorization condition satisfied by its transition probability kernel and then applying a variant of the methodology proposed by S. Datta and W. P. McCormick [Can. J. Stat. 21, No. 2, 181–193 (1993; Zbl 0780.62063)] for bootstrapping additive functionals of the type $$n^{-1} \sum_{n=1}^n f(X_i)$$ 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 $$O_{\mathbb P}(n^{-1})$$ 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 $$O_{\mathbb P}(n^{-1})$$ 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.

### MSC:

 62G09 Nonparametric statistical resampling methods 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 62E20 Asymptotic distribution theory in statistics 65C60 Computational problems in statistics (MSC2010)

Zbl 0780.62063
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### References:

 [1] Asmussen, S. (1987) Applied Probability and Queues. Chichester: Wiley. · Zbl 0624.60098 [2] Athreya, K. and Atuncar, G. (1998) Kernel estimation for real-valued Markov chains. Sankhya Ser. A, 60, 1-17. · Zbl 0977.62093 [3] Athreya, K. and Fuh, C. (1989) Bootstrapping Markov chains: countable case. Technical Report B-89- 7, Institute of statistical Science, Academia Sinica, Taiwan. · Zbl 0765.62078 [4] Bertail, P. and Clémençon, S. (2003) Regenerative block-bootstrap for Markov chains (Revised version). CREST preprint no. 2004-47. http://www.crest.fr/doctravail/document/2004-47.pdf (accessed 10 February 2006). URL: [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. · Zbl 1075.62075 [6] Bertail, P. and Clémençon, S. (2005) Note on the regeneration-based bootstrap for atomic Markov chains. Test. [7] Bickel, P. and Freedman, D. (1981) Some asymptotic theory for the bootstrap. Ann. Statist., 9, 1196- 1217. · Zbl 0449.62034 [8] Bolthausen, E. (1982) The Berry-Esseen theorem for strongly mixing Harris recurrent Markov chains. Z. Wahrscheinlichkeitstheorie Verw. Geb., 60, 283-289. · Zbl 0476.60022 [9] Clémençon, S. (2000) Adaptive estimation of the transition density of a regular Markov chain. Math. Methods Statist., 9, 323-357. · Zbl 1008.62076 [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. · Zbl 1078.60508 [11] Datta, S. and McCormick W. (1993) Regeneration-based bootstrap for Markov chains. Canad. J. Statist., 21, 181-193. JSTOR: · Zbl 0780.62063 [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. · Zbl 1082.60062 [13] Efron, B. (1979) Bootstrap methods: another look at the jackknife. Ann. Statist., 7, 1-26. · Zbl 0406.62024 [14] Franke, J., Kreiss, J.P. and Mammen, E. (2002) Bootstrap of kernel smoothing in nonlinear time series. Bernoulli, 8, 1-37. · Zbl 1006.62038 [15] Götze, F. and Künsch, H. (1996) Second order correctness of the blockwise bootstrap for stationary observations. Ann. Statist., 24, 1914-1933. · Zbl 0906.62040 [16] Hall, P. (1992) The Bootstrap and Edgeworth Expansion. New York: Springer-Verlag. · Zbl 0744.62026 [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. · Zbl 1046.60062 [18] Horowitz, J. (2003) Bootstrap methods for Markov processes. Econometrica, 71, 1049-1082. JSTOR: · Zbl 1154.62361 [19] Jain, J. and Jamison, B. (1967) Contributions to Doeblinś theory of Markov processes. Z. Wahrscheinlichkeitstheorie Verw. Geb., 8, 19-40. · Zbl 0201.50404 [20] Kalashnikov, V. (1978) The Qualitative Analysis of the Behavior of Complex Systems by the Method of Test Functions. Moscow: Nauka. · Zbl 0451.93002 [21] Lahiri, S. (2003) Resampling Methods for Dependent Data. New York: Springer-Verlag. · Zbl 1028.62002 [22] Malinovskii, V. (1987) Limit theorems for Harris Markov chains I. Theory Probab. Appl., 31, 269-285. · Zbl 0657.60087 [23] Malinovskii, V. (1989) Limit theorems for Harris Markov chains II. Theory Probab. Appl., 34, 252-265. · Zbl 0698.60053 [24] Meyn, S. and Tweedie, R. (1996) Markov Chains and Stochastic Stability. London: Springer-Verlag. · Zbl 0925.60001 [25] Nummelin, E. (1978) A splitting technique for Harris recurrent chains. Z. Wahrscheinlichkeitstheorie Verw. Geb., 43, 309-318. · Zbl 0364.60104 [26] Orey, S (1971) Limit Theorems for Markov Chain Transition Probabilities. London: Van Nostrand Reinhold. · Zbl 0295.60054 [27] Rachev, S. and Rüschendorf, L. (1998) Mass Transportation Problems. Vol. II: Applications. New York: Springer-Verlag. · Zbl 0990.60500 [28] Roberts, G. and Rosenthal, J. (1996) Quantitative bounds for convergence rates of continuous time Markov processes. Electron. J. Probab., 9, 1-21. · Zbl 0891.60068 [29] Smith, W. (1955) Regenerative stochastic processes. Proc. Roy. Soc., Lond. Ser. A, 232, 6-31. · Zbl 0067.36301 [30] Thorisson, H. (2000) Coupling, Stationarity, and Regeneration. New York: Springer-Verlag. · Zbl 0949.60007
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