Moderate deviations for stationary sequences of bounded random variables

Jérôme Dedecker; Florence Merlevède; Magda Peligrad; Sergey Utev

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

  • Volume: 45, Issue: 2, page 453-476
  • ISSN: 0246-0203

Abstract

top
In this paper we derive the moderate deviation principle for stationary sequences of bounded random variables under martingale-type conditions. Applications to functions of ϕ-mixing sequences, contracting Markov chains, expanding maps of the interval, and symmetric random walks on the circle are given.

How to cite

top

Dedecker, Jérôme, et al. "Moderate deviations for stationary sequences of bounded random variables." Annales de l'I.H.P. Probabilités et statistiques 45.2 (2009): 453-476. <http://eudml.org/doc/78029>.

@article{Dedecker2009,
abstract = {In this paper we derive the moderate deviation principle for stationary sequences of bounded random variables under martingale-type conditions. Applications to functions of ϕ-mixing sequences, contracting Markov chains, expanding maps of the interval, and symmetric random walks on the circle are given.},
author = {Dedecker, Jérôme, Merlevède, Florence, Peligrad, Magda, Utev, Sergey},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {moderate deviations; martingale approximation; stationary processes},
language = {eng},
number = {2},
pages = {453-476},
publisher = {Gauthier-Villars},
title = {Moderate deviations for stationary sequences of bounded random variables},
url = {http://eudml.org/doc/78029},
volume = {45},
year = {2009},
}

TY - JOUR
AU - Dedecker, Jérôme
AU - Merlevède, Florence
AU - Peligrad, Magda
AU - Utev, Sergey
TI - Moderate deviations for stationary sequences of bounded random variables
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2009
PB - Gauthier-Villars
VL - 45
IS - 2
SP - 453
EP - 476
AB - In this paper we derive the moderate deviation principle for stationary sequences of bounded random variables under martingale-type conditions. Applications to functions of ϕ-mixing sequences, contracting Markov chains, expanding maps of the interval, and symmetric random walks on the circle are given.
LA - eng
KW - moderate deviations; martingale approximation; stationary processes
UR - http://eudml.org/doc/78029
ER -

References

top
  1. [1] H. C. P. Berbee. Random walk with stationary increments and renewal theory. Math. Centre Tracts 112. Mathematisch Centrum, Amsterdam, 1979. Zbl0443.60083MR547109
  2. [2] P. Billingsley. Convergence of Probability Measures. Wiley, New York, 1968. Zbl0172.21201MR233396
  3. [3] R. C. Bradley. Introduction to strong mixing conditions, Volume 1. Technical report, Department of Mathematics, Indiana University, Bloomington. Custom Publishing of I.U., Bloomington, March 2002. 
  4. [4] A. Broise. Transformations dilatantes de l’intervalle et théorèmes limites. Études spectrales d’opérateurs de transfert et applications. Astérisque 238 (1996) 1–109. Zbl0988.37032MR1634271
  5. [5] A. de Acosta and X. Chen. Moderate deviations for empirical measure of Markov chains: upper bound. J. Theoret Probab. 11 (1998) 1075–1110. Zbl0924.60051MR1660920
  6. [6] J. Dedecker and F. Merlevède. Inequalities for partial sums of Hilbert-valued dependent sequences and applications. Math. Methods Statist. 15 (2006) 176–206. MR2256474
  7. [7] J. Dedecker and C. Prieur. An empirical central limit theorem for dependent sequences. Stochastic Process. Appl. 117 (2007) 121–142. Zbl1117.60035MR2287106
  8. [8] J. Dedecker and E. Rio. On mean central limit theorems for stationary sequences. Ann. Inst. H. Poincaré Probab. Statist. 44 (2006), 693–726. Zbl1187.60015MR2446294
  9. [9] B. Delyon, A. Juditsky and R. Liptser. Moderate deviation principle for ergodic Markov chain. Lipschitz summands. In From Stochastic Calculus to Mathematical Finance 189–209. Springer, Berlin, 2006. Zbl1103.60027MR2233540
  10. [10] A. Dembo. Moderate deviations for martingales with bounded jumps. Electron. Comm. Probab. 1 (1996) 11–17. Zbl0854.60027MR1386290
  11. [11] A. Dembo and O. Zeitouni. Moderate deviations of iterates of expanding maps. Statistics and Control of Stochastic Processes 1–11. World Sci. Publi., River Edge, NJ, 1997. Zbl0935.60019MR1647246
  12. [12] A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications, 2nd edition. Springer, New York, 1998. Zbl0896.60013MR1619036
  13. [13] 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
  14. [14] J. D. Deuschel and D. W. Stroock. Large Deviations. Academic Press Inc., Boston, MA, 1989. Zbl0705.60029MR997938
  15. [15] H. Djellout. Moderate deviations for martingale differences and applications to φ-mixing sequences. Stoch. Stoch. Rep. 73 (2002) 37–63. Zbl1005.60044MR1914978
  16. [16] H. Djellout, A. Guillin and L. Wu. Moderate Deviations of empirical periodogram and non-linear functionals of moving average processes. Ann. Inst. H. Poincaré Probab. Statist. 42 (2006), 393–416. Zbl1100.60010MR2242954
  17. [17] F.-Q. Gao. Moderate deviations for martingales and mixing random processes. Stochastic Process. Appl. 61 (1996) 263–275. Zbl0854.60028MR1386176
  18. [18] M. I. Gordin. The central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR 188 (1969) 739–741. Zbl0212.50005MR251785
  19. [19] M. Peligrad and S. Utev. A new maximal inequality and invariance principle for stationary sequences. Ann. Probab. 33 (2005) 798–815. Zbl1070.60025MR2123210
  20. [20] M. Peligrad, S. Utev and W. B. Wu. A maximal Lp-inequality for stationary sequences and its applications. Proc. Amer. Math. Soc. 135 (2007) 541–550. Zbl1107.60011MR2255301
  21. [21] A. Puhalskii. Large deviations of semimartingales via convergence of the predictable characteristics. Stoch. Stoch. Rep. 49 (1994) 27–85. Zbl0827.60017MR1784438
  22. [22] W. M. Schmidt. Diophantine Approximation. Springer, Berlin, 1980. Zbl0421.10019MR568710
  23. [23] L. Wu. Exponential convergence in probability for empirical means of Brownian motion and of random walks. J. Theoret. Probab. 12 (1999) 661–673. Zbl0932.60053MR1702907

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.