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Moderate Deviations for Linear Processes Generated by Martingale-Like Random Variables

Author

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  • Florence Merlevède

    (Université Paris Est-Marne la Vallée, LAMA and C.N.R.S UMR 8050)

  • Magda Peligrad

    (University of Cincinnati)

Abstract

In this paper we study the moderate deviation principle for linear statistics of the type S n =∑i∈Z c n,iξ i , where c n,i are real numbers, and the variables ξ i are in turn stationary martingale differences or dependent sequences satisfying projective criteria. As an application, we obtain the moderate deviation principle and its functional form for sums of a class of linear processes with dependent innovations that might exhibit long memory. A new notion of equivalence of the coefficients allows us to study the difficult case where the variance of S n behaves slower than n. The main tools are: a new type of martingale approximations and moment and maximal inequalities that are important in themselves.

Suggested Citation

  • Florence Merlevède & Magda Peligrad, 2010. "Moderate Deviations for Linear Processes Generated by Martingale-Like Random Variables," Journal of Theoretical Probability, Springer, vol. 23(1), pages 277-300, March.
    • Handle: RePEc:spr:jotpro:v:23:y:2010:i:1:d:10.1007_s10959-009-0218-6
    DOI: 10.1007/s10959-009-0218-6
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    References listed on IDEAS

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    1. Biao Wu, Wei & Min, Wanli, 2005. "On linear processes with dependent innovations," Stochastic Processes and their Applications, Elsevier, vol. 115(6), pages 939-958, June.
    2. Qiying Wang & Yan-Xia Lin & Chandra M. Gulati, 2003. "Strong Approximation for Long Memory Processes with Applications," Journal of Theoretical Probability, Springer, vol. 16(2), pages 377-389, April.
    3. Dong, Zhi-shan & Xi-li, Tan & Yang, Xiao-yun, 2005. "Moderate deviation principles for moving average processes of real stationary sequences," Statistics & Probability Letters, Elsevier, vol. 74(2), pages 139-150, September.
    4. Robinson, Peter M., 1997. "Large-sample inference for nonparametric regression with dependent errors," LSE Research Online Documents on Economics 302, London School of Economics and Political Science, LSE Library.
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    Cited by:

    1. Deng Zhang, 2017. "Tridiagonal Random Matrix: Gaussian Fluctuations and Deviations," Journal of Theoretical Probability, Springer, vol. 30(3), pages 1076-1103, September.

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