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
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topDedecker, 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] H. C. P. Berbee. Random walk with stationary increments and renewal theory. Math. Centre Tracts 112. Mathematisch Centrum, Amsterdam, 1979. Zbl0443.60083MR547109
- [2] P. Billingsley. Convergence of Probability Measures. Wiley, New York, 1968. Zbl0172.21201MR233396
- [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] 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] 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] 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] J. Dedecker and C. Prieur. An empirical central limit theorem for dependent sequences. Stochastic Process. Appl. 117 (2007) 121–142. Zbl1117.60035MR2287106
- [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] 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] A. Dembo. Moderate deviations for martingales with bounded jumps. Electron. Comm. Probab. 1 (1996) 11–17. Zbl0854.60027MR1386290
- [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] A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications, 2nd edition. Springer, New York, 1998. Zbl0896.60013MR1619036
- [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] J. D. Deuschel and D. W. Stroock. Large Deviations. Academic Press Inc., Boston, MA, 1989. Zbl0705.60029MR997938
- [15] H. Djellout. Moderate deviations for martingale differences and applications to φ-mixing sequences. Stoch. Stoch. Rep. 73 (2002) 37–63. Zbl1005.60044MR1914978
- [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] F.-Q. Gao. Moderate deviations for martingales and mixing random processes. Stochastic Process. Appl. 61 (1996) 263–275. Zbl0854.60028MR1386176
- [18] M. I. Gordin. The central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR 188 (1969) 739–741. Zbl0212.50005MR251785
- [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] 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] A. Puhalskii. Large deviations of semimartingales via convergence of the predictable characteristics. Stoch. Stoch. Rep. 49 (1994) 27–85. Zbl0827.60017MR1784438
- [22] W. M. Schmidt. Diophantine Approximation. Springer, Berlin, 1980. Zbl0421.10019MR568710
- [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
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