Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Moderate deviations for stationary sequences of bounded random variables

Jérôme Dedecker, Florence Merlevède, Magda Peligrad, and Sergey Utev

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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.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 45, Number 2 (2009), 453-476.

First available in Project Euclid: 29 April 2009

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Zentralblatt MATH identifier

Primary: 60F10: Large deviations 60G10: Stationary processes

Moderate deviations Martingale approximation Stationary processes


Dedecker, Jérôme; Merlevède, Florence; Peligrad, Magda; Utev, Sergey. Moderate deviations for stationary sequences of bounded random variables. Ann. Inst. H. Poincaré Probab. Statist. 45 (2009), no. 2, 453--476. doi:10.1214/08-AIHP169.

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  • [1] H. C. P. Berbee. Random walk with stationary increments and renewal theory. Math. Centre Tracts 112. Mathematisch Centrum, Amsterdam, 1979.
  • [2] P. Billingsley. Convergence of Probability Measures. Wiley, New York, 1968.
  • [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.
  • [5] A. de Acosta and X. Chen. Moderate deviations for empirical measure of Markov chains: upper bound. J. Theoret Probab. 11 (1998) 1075–1110.
  • [6] J. Dedecker and F. Merlevède. Inequalities for partial sums of Hilbert-valued dependent sequences and applications. Math. Methods Statist. 15 (2006) 176–206.
  • [7] J. Dedecker and C. Prieur. An empirical central limit theorem for dependent sequences. Stochastic Process. Appl. 117 (2007) 121–142.
  • [8] J. Dedecker and E. Rio. On mean central limit theorems for stationary sequences. Ann. Inst. H. Poincaré Probab. Statist. 44 (2006), 693–726.
  • [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.
  • [10] A. Dembo. Moderate deviations for martingales with bounded jumps. Electron. Comm. Probab. 1 (1996) 11–17.
  • [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.
  • [12] A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications, 2nd edition. Springer, New York, 1998.
  • [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.
  • [14] J. D. Deuschel and D. W. Stroock. Large Deviations. Academic Press Inc., Boston, MA, 1989.
  • [15] H. Djellout. Moderate deviations for martingale differences and applications to φ-mixing sequences. Stoch. Stoch. Rep. 73 (2002) 37–63.
  • [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.
  • [17] F.-Q. Gao. Moderate deviations for martingales and mixing random processes. Stochastic Process. Appl. 61 (1996) 263–275.
  • [18] M. I. Gordin. The central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR 188 (1969) 739–741.
  • [19] M. Peligrad and S. Utev. A new maximal inequality and invariance principle for stationary sequences. Ann. Probab. 33 (2005) 798–815.
  • [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.
  • [21] A. Puhalskii. Large deviations of semimartingales via convergence of the predictable characteristics. Stoch. Stoch. Rep. 49 (1994) 27–85.
  • [22] W. M. Schmidt. Diophantine Approximation. Springer, Berlin, 1980.
  • [23] L. Wu. Exponential convergence in probability for empirical means of Brownian motion and of random walks. J. Theoret. Probab. 12 (1999) 661–673.