Issue |
ESAIM: PS
Volume 18, 2014
|
|
---|---|---|
Page(s) | 468 - 482 | |
DOI | https://doi.org/10.1051/ps/2013043 | |
Published online | 08 October 2014 |
Limit theorems for some functionals with heavy tails of a discrete time Markov chain
Institut de Mathématiques de Toulouse. CNRS UMR 5219. Université Paul
Sabatier, 118 route de Narbonne, 31062 Toulouse cedex 09, France
cattiaux@math.univ-toulouse.fr; manoumawaki@gmail.com
Received:
19
December
2012
Revised:
17
May
2013
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 for a possibly non square integrable function
f. 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
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.
Mathematics Subject Classification: 60F05 / 60F17 / 60J05 / 60E07
Key words: Markov chains / stable limit theorems / stable distributions / log-Sobolev inequality / additive functionals / functional limit theorem
© EDP Sciences, SMAI 2014
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.