Limit theorems for some functionals with heavy tails of a discrete time Markov chain

Patrick Cattiaux; Mawaki Manou-Abi

ESAIM: Probability and Statistics (2014)

  • Volume: 18, page 468-482
  • ISSN: 1292-8100

Abstract

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Consider an irreducible, aperiodic and positive recurrent discrete time Markov chain (Xn,n ≥ 0) with invariant distribution μ. We shall investigate the long time behaviour of some functionals of the chain, in particular the additive functional S n = ∑ i = 1 n f ( X i ) S n = ∑ i = 1 n f ( X i ) for a possibly non square integrable functionf. To this end we shall link ergodic properties of the chain to mixing properties, extending known results in the continuous time case. We will then use existing results of convergence to stable distributions, obtained in [M. Denker and A. Jakubowski, Stat. Probab. Lett. 8 (1989) 477–483; M. Tyran-Kaminska, Stochastic Process. Appl. 120 (2010) 1629–1650; D. Krizmanic, Ph.D. thesis (2010); B. Basrak, D. Krizmanic and J. Segers, Ann. Probab. 40 (2012) 2008–2033] for stationary mixing sequences. Contrary to the usual L^2 L 2 framework studied in [P. Cattiaux, D. Chafai and A. Guillin, ALEA, Lat. Am. J. Probab. Math. Stat. 9 (2012) 337–382], where weak forms of ergodicity are sufficient to ensure the validity of the Central Limit Theorem, we will need here strong ergodic properties: the existence of a spectral gap, hyperboundedness (or hypercontractivity). These properties are also discussed. Finally we give explicit examples.

How to cite

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Cattiaux, Patrick, and Manou-Abi, Mawaki. "Limit theorems for some functionals with heavy tails of a discrete time Markov chain." ESAIM: Probability and Statistics 18 (2014): 468-482. <http://eudml.org/doc/274363>.

@article{Cattiaux2014,
abstract = {Consider an irreducible, aperiodic and positive recurrent discrete time Markov chain (Xn,n ≥ 0) with invariant distribution μ. We shall investigate the long time behaviour of some functionals of the chain, in particular the additive functional $S_\{n\}=\sum _\{i=1\}^\{n\}f(X_\{i\})$ S n = ∑ i = 1 n f ( X i ) for a possibly non square integrable functionf. To this end we shall link ergodic properties of the chain to mixing properties, extending known results in the continuous time case. We will then use existing results of convergence to stable distributions, obtained in [M. Denker and A. Jakubowski, Stat. Probab. Lett. 8 (1989) 477–483; M. Tyran-Kaminska, Stochastic Process. Appl. 120 (2010) 1629–1650; D. Krizmanic, Ph.D. thesis (2010); B. Basrak, D. Krizmanic and J. Segers, Ann. Probab. 40 (2012) 2008–2033] for stationary mixing sequences. Contrary to the usual L^2 L 2 framework studied in [P. Cattiaux, D. Chafai and A. Guillin, ALEA, Lat. Am. J. Probab. Math. Stat. 9 (2012) 337–382], where weak forms of ergodicity are sufficient to ensure the validity of the Central Limit Theorem, we will need here strong ergodic properties: the existence of a spectral gap, hyperboundedness (or hypercontractivity). These properties are also discussed. Finally we give explicit examples.},
author = {Cattiaux, Patrick, Manou-Abi, Mawaki},
journal = {ESAIM: Probability and Statistics},
keywords = {Markov chains; stable limit theorems; stable distributions; log-Sobolev inequality; additive functionals; functional limit theorem; functional limit theorems},
language = {eng},
pages = {468-482},
publisher = {EDP-Sciences},
title = {Limit theorems for some functionals with heavy tails of a discrete time Markov chain},
url = {http://eudml.org/doc/274363},
volume = {18},
year = {2014},
}

TY - JOUR
AU - Cattiaux, Patrick
AU - Manou-Abi, Mawaki
TI - Limit theorems for some functionals with heavy tails of a discrete time Markov chain
JO - ESAIM: Probability and Statistics
PY - 2014
PB - EDP-Sciences
VL - 18
SP - 468
EP - 482
AB - Consider an irreducible, aperiodic and positive recurrent discrete time Markov chain (Xn,n ≥ 0) with invariant distribution μ. We shall investigate the long time behaviour of some functionals of the chain, in particular the additive functional $S_{n}=\sum _{i=1}^{n}f(X_{i})$ S n = ∑ i = 1 n f ( X i ) for a possibly non square integrable functionf. To this end we shall link ergodic properties of the chain to mixing properties, extending known results in the continuous time case. We will then use existing results of convergence to stable distributions, obtained in [M. Denker and A. Jakubowski, Stat. Probab. Lett. 8 (1989) 477–483; M. Tyran-Kaminska, Stochastic Process. Appl. 120 (2010) 1629–1650; D. Krizmanic, Ph.D. thesis (2010); B. Basrak, D. Krizmanic and J. Segers, Ann. Probab. 40 (2012) 2008–2033] for stationary mixing sequences. Contrary to the usual L^2 L 2 framework studied in [P. Cattiaux, D. Chafai and A. Guillin, ALEA, Lat. Am. J. Probab. Math. Stat. 9 (2012) 337–382], where weak forms of ergodicity are sufficient to ensure the validity of the Central Limit Theorem, we will need here strong ergodic properties: the existence of a spectral gap, hyperboundedness (or hypercontractivity). These properties are also discussed. Finally we give explicit examples.
LA - eng
KW - Markov chains; stable limit theorems; stable distributions; log-Sobolev inequality; additive functionals; functional limit theorem; functional limit theorems
UR - http://eudml.org/doc/274363
ER -

References

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