The search session has expired. Please query the service again.

Pointwise ergodic theorems with rate and application to the CLT for Markov chains

Christophe Cuny; Michael Lin

Annales de l'I.H.P. Probabilités et statistiques (2009)

  • Volume: 45, Issue: 3, page 710-733
  • ISSN: 0246-0203

Abstract

top
Let T be Dunford–Schwartz operator on a probability space (Ω, μ). For f∈Lp(μ), p>1, we obtain growth conditions on ‖∑k=1nTkf‖p which imply that (1/n1/p)∑k=1nTkf→0 μ-a.e. In the particular case that p=2 and T is the isometry induced by a probability preserving transformation we get better results than in the general case; these are used to obtain a quenched central limit theorem for additive functionals of stationary ergodic Markov chains, which improves those of Derriennic–Lin and Wu–Woodroofe.

How to cite

top

Cuny, Christophe, and Lin, Michael. "Pointwise ergodic theorems with rate and application to the CLT for Markov chains." Annales de l'I.H.P. Probabilités et statistiques 45.3 (2009): 710-733. <http://eudml.org/doc/78040>.

@article{Cuny2009,
abstract = {Let T be Dunford–Schwartz operator on a probability space (Ω, μ). For f∈Lp(μ), p&gt;1, we obtain growth conditions on ‖∑k=1nTkf‖p which imply that (1/n1/p)∑k=1nTkf→0 μ-a.e. In the particular case that p=2 and T is the isometry induced by a probability preserving transformation we get better results than in the general case; these are used to obtain a quenched central limit theorem for additive functionals of stationary ergodic Markov chains, which improves those of Derriennic–Lin and Wu–Woodroofe.},
author = {Cuny, Christophe, Lin, Michael},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {ergodic theorems with rates; central limit theorem for Markov chains; Dunford–Schwartz operators; probability preserving transformations; Dunford-Schwartz operators},
language = {eng},
number = {3},
pages = {710-733},
publisher = {Gauthier-Villars},
title = {Pointwise ergodic theorems with rate and application to the CLT for Markov chains},
url = {http://eudml.org/doc/78040},
volume = {45},
year = {2009},
}

TY - JOUR
AU - Cuny, Christophe
AU - Lin, Michael
TI - Pointwise ergodic theorems with rate and application to the CLT for Markov chains
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2009
PB - Gauthier-Villars
VL - 45
IS - 3
SP - 710
EP - 733
AB - Let T be Dunford–Schwartz operator on a probability space (Ω, μ). For f∈Lp(μ), p&gt;1, we obtain growth conditions on ‖∑k=1nTkf‖p which imply that (1/n1/p)∑k=1nTkf→0 μ-a.e. In the particular case that p=2 and T is the isometry induced by a probability preserving transformation we get better results than in the general case; these are used to obtain a quenched central limit theorem for additive functionals of stationary ergodic Markov chains, which improves those of Derriennic–Lin and Wu–Woodroofe.
LA - eng
KW - ergodic theorems with rates; central limit theorem for Markov chains; Dunford–Schwartz operators; probability preserving transformations; Dunford-Schwartz operators
UR - http://eudml.org/doc/78040
ER -

References

top
  1. [1] I. Assani and M. Lin. On the one-sided ergodic Hilbert transform. Contemp. Math. 430 (2007) 20–39. Zbl1134.47007MR2331323
  2. [2] G. Cohen and M. Lin. Extensions of the Menchoff–Rademacher theorem with applications to ergodic theory. Israel J. Math. 148 (2005) 41–86. Zbl1086.60019MR2191224
  3. [3] J. Dedecker and E. Rio. On the functional central limit theorem for stationary processes. Ann. Inst. H. Poincaré Probab. Statist. 36 (2000) 1–34. Zbl0949.60049MR1743095
  4. [4] Y. Derriennic. Some aspects of recent works on limit theorems in ergodic theory with special emphasis on the “central limit theorem.” Discrete Contin. Dyn. Syst. 15 (2006) 143–158. Zbl1107.37009MR2191389
  5. [5] Y. Derriennic and M. Lin. Fractional Poisson equations and ergodic theorems for fractional coboundaries. Israel J. Math. 123 (2001) 93–130. Zbl0988.47009MR1835290
  6. [6] Y. Derriennic and M. Lin. The central limit theorem for Markov chains with normal transition operators, started at a point. Probab. Theory Related Fields 119 (2001) 508–528. Zbl0974.60017MR1826405
  7. [7] Y. Derriennic and M. Lin. The central limit theorem for Markov chains started at a point. Probab. Theory Related Fields 125 (2003) 73–76. Zbl1012.60028MR1952457
  8. [8] Y. Derriennic and M. Lin. The central limit theorem for random walks on orbits of probability preserving transformations. Contemp. Math. 444 (2007) 31–51. Zbl1130.60026MR2423622
  9. [9] N. Dunford and J. Schwartz. Linear Operators, Part I. Wiley, New York. 1958. Zbl0084.10402MR1009162
  10. [10] N. Dunford and J. Schwartz. Linear Operators, Part II. Wiley, New York, 1963. Zbl0128.34803MR1009163
  11. [11] W. Feller. An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edition. Wiley, New York, 1971. Zbl0219.60003MR270403
  12. [12] V. F. Gaposhkin. On the dependence of the convergence rate in the SLLN for stationary processes on the rate of decay of correlation function. Theory Probab. Appl. 26 (1981) 706–720. Zbl0488.60040MR636767
  13. [13] V. F. Gaposhkin. Spectral criteria for the existence of generalized ergodic transformations (in Russian). Teor. Veroyatnost. i Primenen. 41(2) (1996) 251–271. (Translation in Theory Probab. Appl. 41 (1996) 247–264 (1997).) Zbl0881.60038MR1445750
  14. [14] M. Gordin and B. Lifshitz. A central limit theorem for Markov process. Soviet Math. Doklady 19 (1978) 392–394. Zbl0395.60057
  15. [15] M. Gordin and B. Lifshitz. A remark about a Markov process with normal transition operator. Proc. Third Vilnius Conf. Probab. Statist. 147–148. Akad. Nauk Litovsk., Vilnius, 1981 (in Russian). 
  16. [16] M. Gordin and B. Lifshitz. The central limit theorem for Markov processes with normal transition operator, and a strong form of the central limit theorem. In Limit Theorems for Functionals of Random Walks Sections IV.7 and IV.8. A. Borodin and I. Ibragimov (Eds). Proc. Steklov Inst. Math. 195, 1994. (English translation Amer. Math. Soc., Providence, RI, 1995.) 
  17. [17] A. G. Kachurovskii. The rate of convergence in ergodic theorems. Russian Math. Surveys 51 (1996) 653–703. Zbl0880.60024MR1422228
  18. [18] C. Kipnis and S. R. Varadhan. Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104 (1986) 1–19. Zbl0588.60058MR834478
  19. [19] U. Krengel. Ergodic Theorems. De Gruyter, Berlin, 1985. Zbl0575.28009MR797411
  20. [20] M. Maxwell and M. Woodroofe. Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28 (2000) 713–724. Zbl1044.60014MR1782272
  21. [21] F. Moricz. Moment inequalities and the strong laws of large numbers. Z. Wahrsch. Verw. Gebiete 35 (1976) 299–314. Zbl0314.60023MR407950
  22. [22] M. Peligrad and S. Utev. A new maximal inequality and invariance principle for stationary sequences. Ann. Probab. 33 (2005) 798–815. Zbl1070.60025MR2123210
  23. [23] F. Rassoul-Agha and T. Seppäläinen. An almost sure invariance principle for additive functionals of Markov chains. Statist. Probab. Lett. 78 (2008) 854–860. Zbl1139.60317MR2398359
  24. [24] F. Riesz and B. Sz.-Nagy. Leçons D’analyse Fonctionnelle, 3rd edition. Akadémiai Kiadó, Budapest, 1955. Zbl0064.35404
  25. [25] M. Weber. Uniform bounds under increment conditions. Trans. Amer. Math. Soc. 358 (2006) 911–936. Zbl1078.60025MR2177045
  26. [26] W. B. Wu. Strong invariance principles for dependent random variables. Ann. Probab. 35 (2007) 2294–2320. Zbl1166.60307MR2353389
  27. [27] W. B. Wu and M. Woodroofe. Martingale approximations for sums of stationary processes. Ann. Probab. 32 (2004) 1674–1690. Zbl1057.60022MR2060314
  28. [28] O. Zhao and M. Woodroofe. Laws of the iterated logarithm for stationary processes. Ann. Probab. 36 (2008) 127–142. Zbl1130.60039MR2370600
  29. [29] A. Zygmund. Trigonometric Series, corrected 2nd edition. Cambridge Univ. Press, Cambridge, UK, 1969. Zbl0367.42001

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.