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Bernstein inequality and moderate deviations under strong mixing conditions

Florence Merlevède, Magda Peligrad, and Emmanuel Rio

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Abstract

In this paper we obtain a Bernstein type inequality for a class of weakly dependent and bounded random variables. The proofs lead to a moderate deviations principle for sums of bounded random variables with exponential decay of the strong mixing coefficients that complements the large deviation result obtained by Bryc and Dembo (1998) under superexponential mixing rates.

Chapter information

Source
Christian Houdré, Vladimir Koltchinskii, David M. Mason and Magda Peligrad, eds., High Dimensional Probability V: The Luminy Volume (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2009), 273-292

Dates
First available in Project Euclid: 2 February 2010

Permanent link to this document
https://projecteuclid.org/euclid.imsc/1265119274

Digital Object Identifier
doi:10.1214/09-IMSCOLL518

Mathematical Reviews number (MathSciNet)
MR2797953

Zentralblatt MATH identifier
1243.60019

Subjects
Primary: 60E15: Inequalities; stochastic orderings 60F10: Large deviations 62G07: Density estimation

Keywords
deviation inequality moderate deviations principle weakly dependent sequences strong mixing

Rights
Copyright © 2009, Institute of Mathematical Statistics

Citation

Merlevède, Florence; Peligrad, Magda; Rio, Emmanuel. Bernstein inequality and moderate deviations under strong mixing conditions. High Dimensional Probability V: The Luminy Volume, 273--292, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2009. doi:10.1214/09-IMSCOLL518. https://projecteuclid.org/euclid.imsc/1265119274


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References

  • [1] Adamczak, R. (2008). A tail inequality for suprema of unbounded empirical processes with applications to Markov chains. Electron. J. Probab. 13 1000–1034.
  • [2] Arcones, M. A. (2003). Moderate deviations of empirical processes. In Stochastic Inequalities and Applications. Progr. Probab. 56 189-212. Birkhäuser, Basel.
  • [3] Bosq, D. (1998). Nonparametric Statistics for Stochastic Processes. Estimation and Prediction, 2nd ed. Lecture Notes in Statistics. Springer-Verlag.
  • [4] Bradley, R. C. (1997). On quantiles and the central limit question for strongly mixing sequences. J. Theor. Probab. 10 507–555.
  • [5] Bradley, R. C. (2007). Introduction to Strong Mixing Conditions, Vol. 1, 2, 3. Kendrick Press.
  • [6] Bryc, W. and Dembo, A. (1996). Large deviations and strong mixing. Ann. Inst. Henri Poincaré 32 549–569.
  • [7] Castellana, J. V. and Leadbetter, M. R. (1986). On smoothed probability density estimation for stationary process. Stochastic Process. Appl. 21 179–193.
  • [8] Chen, X. and de Acosta, A. (1998). Moderate deviations for empirical measures of Markov chains: Upper bounds. J. Theoret. Probab. 11(4) 1075–1110.
  • [9] de Acosta, A. (1997). Moderate deviations for empirical measures of Markov chains: Lower bounds. Ann. Probab. 25(1) 259–284.
  • [10] Dedecker, J. and Prieur, C. (2004). Coupling for tau-dependent sequences and applications. J. Theoret. Probab. 17 861–885.
  • [11] Dedecker, J., Merlevède, F., Peligrad, M. and Utev, S. (2009). Moderate deviations for stationary sequences of bounded random variables. Ann. Inst. H. Poincaré Probab. Statist. 45 453–476.
  • [12] Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Springer, New York.
  • [13] Doukhan, P. and Neumann, M. (2007). Probability and moment inequalities for sums of weakly dependent random variables, with applications. Stochastic Process. Appl. 117(7) 878–903.
  • [14] Ibragimov, I. A. (1962). Some limit theorems for stationary processes. Theor. Probab. Appl. 7 349–382.
  • [15] Leblanc, F. (1997). Density estimation for a class of continuous time processes. Math. Methods of Stat. 6 171–199.
  • [16] Merlevède, F. and Peligrad, M. (2009). Functional moderate deviations for triangular arrays and applications. ALEA. 5 3–20.
  • [17] Merlevède, F., Peligrad, M. and Rio, E. (2009). A Bernstein type inequality and moderate deviations for weakly dependent sequences. Preprint. Available at arXiv: 0902.0582.
  • [18] Rio, E. (2000). Théorie asymptotique des processus aléatoires faiblement dépendants. Mathématiques et Applications 31. Springer, Berlin.
  • [19] Rosenblatt, M. (1956). A central limit theorem and a strong mixing condition. Proc. Nat. Acad. Sci. U. S. A. 42 43–47.
  • [20] Veretennikov, A. Yu. (1990). On hypoellipticity conditions and estimates of the mixing rate for stochastic differential equations. Soviet Math. Dokl. 40 94–97.
  • [21] Tsirelson, B. (2008). Moderate deviations for random fields and random complex zeroes. Available at arXiv:0801.1050.