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On the Weak Invariance Principle for Non-Adapted Sequences under Projective Criteria

Author

Listed:
  • Jérôme Dedecker

    (Université Paris VI)

  • Florence Merlevède

    (Université Paris VI, et C.N.R.S UMR 7599)

  • Dalibor Volný

    (Université de Rouen)

Abstract

In this paper we study the central limit theorem and its weak invariance principle for sums of non-adapted stationary sequences, under different normalizations. Our conditions involve the conditional expectation of the variables with respect to a given σ-algebra, as done in Gordin (Dokl. Akad. Nauk SSSR 188, 739–741, 1969) and Heyde (Z. Wahrsch. verw. Gebiete 30, 315–320, 1974). These conditions are well adapted to a large variety of examples, including linear processes with dependent innovations or regular functions of linear processes.

Suggested Citation

  • Jérôme Dedecker & Florence Merlevède & Dalibor Volný, 2007. "On the Weak Invariance Principle for Non-Adapted Sequences under Projective Criteria," Journal of Theoretical Probability, Springer, vol. 20(4), pages 971-1004, December.
    • Handle: RePEc:spr:jotpro:v:20:y:2007:i:4:d:10.1007_s10959-007-0090-1
    DOI: 10.1007/s10959-007-0090-1
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    References listed on IDEAS

    as
    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. Woodroofe, Michael, 1992. "A central limit theorem for functions of a Markov chain with applications to shifts," Stochastic Processes and their Applications, Elsevier, vol. 41(1), pages 33-44, May.
    3. Dedecker, Jérôme & Merlevède, Florence, 2003. "The conditional central limit theorem in Hilbert spaces," Stochastic Processes and their Applications, Elsevier, vol. 108(2), pages 229-262, December.
    4. Volný, Dalibor, 1993. "Approximating martingales and the central limit theorem for strictly stationary processes," Stochastic Processes and their Applications, Elsevier, vol. 44(1), pages 41-74, January.
    5. Hannan, E. J., 1979. "The central limit theorem for time series regression," Stochastic Processes and their Applications, Elsevier, vol. 9(3), pages 281-289, December.
    6. Wang, Qiying & Lin, Yan-Xia & Gulati, Chandra M., 2002. "The Invariance Principle For Linear Processes With Applications," Econometric Theory, Cambridge University Press, vol. 18(1), pages 119-139, February.
    7. Florence Merlevède & Magda Peligrad, 2006. "On the Weak Invariance Principle for Stationary Sequences under Projective Criteria," Journal of Theoretical Probability, Springer, vol. 19(3), pages 647-689, December.
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    Cited by:

    1. Yizao Wang, 2014. "An Invariance Principle for Fractional Brownian Sheets," Journal of Theoretical Probability, Springer, vol. 27(4), pages 1124-1139, December.

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