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.
Dans cet article, nous établissons un principe de déviation modérée pour des suites stationnaires de variables aléatoires bornées sous différentes conditions projectives. Nous appliquons ces résultats aux suites ϕ-mélangeantes, à certaines chaînes de Markov contractantes, aux transformations uniformément dilatantes de l'intervalle, ainsi qu'à la marche aléatoire symétrique sur le cercle.
Keywords: moderate deviations, martingale approximation, stationary processes
@article{AIHPB_2009__45_2_453_0, author = {Dedecker, J\'er\^ome and Merlev\`ede, Florence and Peligrad, Magda and Utev, Sergey}, title = {Moderate deviations for stationary sequences of bounded random variables}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {453--476}, publisher = {Gauthier-Villars}, volume = {45}, number = {2}, year = {2009}, doi = {10.1214/08-AIHP169}, mrnumber = {2521409}, zbl = {1172.60005}, language = {en}, url = {http://archive.numdam.org/articles/10.1214/08-AIHP169/} }
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 SP - 453 EP - 476 VL - 45 IS - 2 PB - Gauthier-Villars UR - http://archive.numdam.org/articles/10.1214/08-AIHP169/ DO - 10.1214/08-AIHP169 LA - en ID - AIHPB_2009__45_2_453_0 ER -
%0 Journal Article %A Dedecker, Jérôme %A Merlevède, Florence %A Peligrad, Magda %A Utev, Sergey %T Moderate deviations for stationary sequences of bounded random variables %J Annales de l'I.H.P. Probabilités et statistiques %D 2009 %P 453-476 %V 45 %N 2 %I Gauthier-Villars %U http://archive.numdam.org/articles/10.1214/08-AIHP169/ %R 10.1214/08-AIHP169 %G en %F AIHPB_2009__45_2_453_0
Dedecker, Jérôme; Merlevède, Florence; Peligrad, Magda; Utev, Sergey. Moderate deviations for stationary sequences of bounded random variables. Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 2, pp. 453-476. doi : 10.1214/08-AIHP169. http://archive.numdam.org/articles/10.1214/08-AIHP169/
[1] Random walk with stationary increments and renewal theory. Math. Centre Tracts 112. Mathematisch Centrum, Amsterdam, 1979. | MR | Zbl
.[2] Convergence of Probability Measures. Wiley, New York, 1968. | MR | Zbl
.[3] Introduction to strong mixing conditions, Volume 1. Technical report, Department of Mathematics, Indiana University, Bloomington. Custom Publishing of I.U., Bloomington, March 2002.
.[4] Transformations dilatantes de l'intervalle et théorèmes limites. Études spectrales d'opérateurs de transfert et applications. Astérisque 238 (1996) 1-109. | MR | Zbl
.[5] Moderate deviations for empirical measure of Markov chains: upper bound. J. Theoret Probab. 11 (1998) 1075-1110. | MR | Zbl
and .[6] Inequalities for partial sums of Hilbert-valued dependent sequences and applications. Math. Methods Statist. 15 (2006) 176-206. | MR
and .[7] An empirical central limit theorem for dependent sequences. Stochastic Process. Appl. 117 (2007) 121-142. | MR | Zbl
and .[8] On mean central limit theorems for stationary sequences. Ann. Inst. H. Poincaré Probab. Statist. 44 (2006), 693-726. | Numdam | MR
and .[9] Moderate deviation principle for ergodic Markov chain. Lipschitz summands. In From Stochastic Calculus to Mathematical Finance 189-209. Springer, Berlin, 2006. | MR | Zbl
, and .[10] Moderate deviations for martingales with bounded jumps. Electron. Comm. Probab. 1 (1996) 11-17. | MR | Zbl
.[11] Moderate deviations of iterates of expanding maps. Statistics and Control of Stochastic Processes 1-11. World Sci. Publi., River Edge, NJ, 1997. | MR | Zbl
and .[12] Large Deviations Techniques and Applications, 2nd edition. Springer, New York, 1998. | MR | Zbl
and .[13] The central limit theorem for Markov chains with normal transition operators, started at a point. Probab. Theory Related Fields 119 (2001) 508-528. | MR | Zbl
and .[14] Large Deviations. Academic Press Inc., Boston, MA, 1989. | MR | Zbl
and .[15] Moderate deviations for martingale differences and applications to φ-mixing sequences. Stoch. Stoch. Rep. 73 (2002) 37-63. | MR | Zbl
.[16] Moderate Deviations of empirical periodogram and non-linear functionals of moving average processes. Ann. Inst. H. Poincaré Probab. Statist. 42 (2006), 393-416. | Numdam | MR | Zbl
, and .[17] Moderate deviations for martingales and mixing random processes. Stochastic Process. Appl. 61 (1996) 263-275. | MR | Zbl
.[18] The central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR 188 (1969) 739-741. | MR | Zbl
.[19] A new maximal inequality and invariance principle for stationary sequences. Ann. Probab. 33 (2005) 798-815. | MR | Zbl
and .[20] A maximal Lp-inequality for stationary sequences and its applications. Proc. Amer. Math. Soc. 135 (2007) 541-550. | MR | Zbl
, and .[21] Large deviations of semimartingales via convergence of the predictable characteristics. Stoch. Stoch. Rep. 49 (1994) 27-85. | MR | Zbl
.[22] Diophantine Approximation. Springer, Berlin, 1980. | MR | Zbl
.[23] Exponential convergence in probability for empirical means of Brownian motion and of random walks. J. Theoret. Probab. 12 (1999) 661-673. | MR | Zbl
.Cited by Sources: