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Functional Limit Theorem for the Empirical Process of a Class of Bernoulli Shifts with Long Memory

Author

Listed:
  • Paul Doukhan

    (LS-CREST and University Cergy Pontoise)

  • Gabriel Lang

    (Laboratoire GRESE)

  • Donatas Surgailis

    (Vilnius Institute of Mathematics and Informatics)

  • Marie-Claude Viano

    (Laboratoire de Mathématiques Appliquées)

Abstract

We prove a functional central limit theorem for the empirical process of a stationary process X t =Y t +V t , where Y t is a long memory moving average in i.i.d. r.v.’s ζ s , s ≤ t, and V t =V (ζ t , ζt-1,...) is a weakly dependent nonlinear Bernoulli shift. Conditions of weak dependence of V t are written in terms of L2-norms of shift-cut differences V (ζ t , ζt-n, 0,...,) − V(ζ t ,...,ζt-n+1, 0,...). Examples of Bernoulli shifts are discussed. The limit empirical process is a degenerated process of the form f(x)Z, where f is the marginal p.d.f. of X0 and Z is a standard normal r.v. The proof is based on a uniform reduction principle for the empirical process.

Suggested Citation

  • Paul Doukhan & Gabriel Lang & Donatas Surgailis & Marie-Claude Viano, 2005. "Functional Limit Theorem for the Empirical Process of a Class of Bernoulli Shifts with Long Memory," Journal of Theoretical Probability, Springer, vol. 18(1), pages 161-186, January.
    • Handle: RePEc:spr:jotpro:v:18:y:2005:i:1:d:10.1007_s10959-004-2593-3
    DOI: 10.1007/s10959-004-2593-3
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    References listed on IDEAS

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    1. Giraitis, Liudas & Koul, Hira L. & Surgailis, Donatas, 1996. "Asymptotic normality of regression estimators with long memory errors," Statistics & Probability Letters, Elsevier, vol. 29(4), pages 317-335, September.
    2. Robinson, P. M., 1991. "Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression," Journal of Econometrics, Elsevier, vol. 47(1), pages 67-84, January.
    3. Giraitis, Liudas & Kokoszka, Piotr & Leipus, Remigijus, 2000. "Stationary Arch Models: Dependence Structure And Central Limit Theorem," Econometric Theory, Cambridge University Press, vol. 16(1), pages 3-22, February.
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