Bounds on regeneration times and limit theorems for subgeometric Markov chains

Randal Douc; Arnaud Guillin; Eric Moulines

Annales de l'I.H.P. Probabilités et statistiques (2008)

  • Volume: 44, Issue: 2, page 239-257
  • ISSN: 0246-0203

Abstract

top
This paper studies limit theorems for Markov chains with general state space under conditions which imply subgeometric ergodicity. We obtain a central limit theorem and moderate deviation principles for additive not necessarily bounded functional of the Markov chains under drift and minorization conditions which are weaker than the Foster–Lyapunov conditions. The regeneration-split chain method and a precise control of the modulated moment of the hitting time to small sets are employed in the proof.

How to cite

top

Douc, Randal, Guillin, Arnaud, and Moulines, Eric. "Bounds on regeneration times and limit theorems for subgeometric Markov chains." Annales de l'I.H.P. Probabilités et statistiques 44.2 (2008): 239-257. <http://eudml.org/doc/77968>.

@article{Douc2008,
abstract = {This paper studies limit theorems for Markov chains with general state space under conditions which imply subgeometric ergodicity. We obtain a central limit theorem and moderate deviation principles for additive not necessarily bounded functional of the Markov chains under drift and minorization conditions which are weaker than the Foster–Lyapunov conditions. The regeneration-split chain method and a precise control of the modulated moment of the hitting time to small sets are employed in the proof.},
author = {Douc, Randal, Guillin, Arnaud, Moulines, Eric},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {stochastic monotonicity; rates of convergence; Markov chains},
language = {eng},
number = {2},
pages = {239-257},
publisher = {Gauthier-Villars},
title = {Bounds on regeneration times and limit theorems for subgeometric Markov chains},
url = {http://eudml.org/doc/77968},
volume = {44},
year = {2008},
}

TY - JOUR
AU - Douc, Randal
AU - Guillin, Arnaud
AU - Moulines, Eric
TI - Bounds on regeneration times and limit theorems for subgeometric Markov chains
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2008
PB - Gauthier-Villars
VL - 44
IS - 2
SP - 239
EP - 257
AB - This paper studies limit theorems for Markov chains with general state space under conditions which imply subgeometric ergodicity. We obtain a central limit theorem and moderate deviation principles for additive not necessarily bounded functional of the Markov chains under drift and minorization conditions which are weaker than the Foster–Lyapunov conditions. The regeneration-split chain method and a precise control of the modulated moment of the hitting time to small sets are employed in the proof.
LA - eng
KW - stochastic monotonicity; rates of convergence; Markov chains
UR - http://eudml.org/doc/77968
ER -

References

top
  1. [1] E. Bolthausen. The Berry–Esseén theorem for strongly mixing Harris recurrent Markov chains. Z. Wahrsch. Verw. Gebiete 60 (1982) 283–289. Zbl0476.60022MR664418
  2. [2] X. Chen. Moderate deviations for m-dependent random variables with Banach space values. Statist. Probab. Lett. 35 (1997) 123–134. Zbl0887.60010MR1483265
  3. [3] X. Chen. Limit theorems for functionals of ergodic Markov chains with general state space. Mem. Amer. Math. Soc. 139 (1999) xiv + 203. Zbl0952.60014MR1491814
  4. [4] S. J. M. Clémençon. Moment and probability inequalities for sums of bounded additive functionals of regular Markov chains via the Nummelin splitting technique. Statist. Probab. Lett. 55 (2001) 227–238. Zbl1078.60508MR1867526
  5. [5] A. de Acosta. Moderate deviations for empirical measures of Markov chains: lower bounds. Ann. Probab. 25 (1997) 259–284. Zbl0877.60019MR1428509
  6. [6] A. de Acosta and X. Chen. Moderate deviations for empirical measures of Markov chains: upper bounds. J. Theoret. Probab. 11 (1998) 1075–1110. Zbl0924.60051MR1660920
  7. [7] H. Djellout and A. Guillin. Moderate deviations for Markov chains with atom. Stochastic Process. Appl. 95 (2001) 203–217. Zbl1059.60029MR1854025
  8. [8] R. Douc, G. Fort, E. Moulines and P. Soulier. Practical drift conditions for subgeometric rates of convergence. Ann. Appl. Probab. 14 (2004) 1353–1377. Zbl1082.60062MR2071426
  9. [9] G. Fort and E. Moulines. V-subgeometric ergodicity for a Hastings–Metropolis algorithm. Statist. Probab. Lett. 49 (2000) 401–410. Zbl0981.60032MR1796485
  10. [10] D. H. Fuk and S. V. Nagaev. Probabilistic inequalities for sums of independent random variables. Teor. Verojatnost. i Primenen. 16 (1971) 660–675. Zbl0259.60024MR293695
  11. [11] S. F. Jarner and G. O. Roberts. Polynomial convergence rates of Markov chains. Ann. Appl. Probab. 12 (2002) 224–247. Zbl1012.60062MR1890063
  12. [12] G. Jones and J. Hobert. Honest exploration of intractable probability distributions via Markov chain Monte Carlo. Statist. Sci. 16 (2001) 312–334. Zbl1127.60309MR1888447
  13. [13] M. Ledoux. Sur les déviations modérées des sommes de variables aléatoires vectorielles indépendantes de même loi. Ann. Inst. H. Poincaré Probab. Statist. 28 (1992) 267–280. Zbl0751.60009MR1162575
  14. [14] S. P. Meyn and R. L. Tweedie. Markov Chains and Stochastic Stability. Springer, London, 1993. Zbl0925.60001MR1287609
  15. [15] E. Nummelin. General Irreducible Markov Chains and Non-Negative Operators. Cambridge University Press, 1984. Zbl0551.60066MR776608
  16. [16] E. Nummelin and P. Tuominen. The rate of convergence in Orey’s theorem for Harris recurrent Markov chains with applications to renewal theory. Stochastic Process. Appl. 15 (1983) 295–311. Zbl0532.60060MR711187
  17. [17] G. O. Roberts and R. L. Tweedie. Bounds on regeneration times and convergence rates for Markov chains. Stochastic Process. Appl. 80 (1999) 211–229. Zbl0961.60066MR1682243
  18. [18] P. Tuominen and R. Tweedie. Subgeometric rates of convergence of f-ergodic Markov chains. Adv. in Appl. Probab. 26 (1994) 775–798. Zbl0803.60061MR1285459

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.