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Central limit theorems for nonlinear functionals of stationary Gaussian processes
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  • Published: September 1989

Central limit theorems for nonlinear functionals of stationary Gaussian processes

  • Daniel Chambers1 &
  • Eric Slud2 

Probability Theory and Related Fields volume 80, pages 323–346 (1989)Cite this article

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  • 46 Citations

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Summary

Let X=(X t ,t∈ℝ) be a stationary Gaussian process on (Ω, ℱ,P), letH(X) be the Hilbert space of variables inL 2 (Ω,P) which are measurable with respect toX, and let (U s ,s∈ℝ) be the associated family of time-shift operators. We sayY∈H(X) (withE(Y)=0) satisfies the functional central limit theorem or FCLT [respectively, the central limit theorem of CLT if

in

[respectively,

], where

$$Y_T (t) \equiv {{\int\limits_0^{Tt} {U_s \circ Yds} } \mathord{\left/ {\vphantom {{\int\limits_0^{Tt} {U_s \circ Yds} } {\left\{ {Var\left( {\int\limits_0^T {U_s \circ Yds} } \right)} \right\}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}} \right. \kern-\nulldelimiterspace} {\left\{ {Var\left( {\int\limits_0^T {U_s \circ Yds} } \right)} \right\}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}$$

andW(•) is a standard Wiener process on [0,1]. This paper provides some general sufficient conditions onX andY ensuring thatY satisfies the CLT or FCLT. Examples ofY are given which satisfy the CLT but not the FCLT. This work extends CLT's of Maruyama (1976) and Breuer and Major (1983).

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References

  1. Billingsley, P.: Convergence of probability measures. New York: Wiley 1968

    Google Scholar 

  2. Billingsley, P.: Probability and measure. New York: Wiley 1979

    Google Scholar 

  3. Breuer, P., Major, P.: Central limit theorems for non-linear functionals of Gaussian fields. J. Multivar. Anal.13, 425–441 (1983)

    Google Scholar 

  4. Cuzick, J.: A central limit theorem for the number of zeroes of a stationary Gaussian process. Ann. Probab.4, 547–556 (1976)

    Google Scholar 

  5. Dobrushin, R.: Gaussian and their subordinated self-similar random generalized fields. Ann. Probab.7, 1–28 (1979)

    Google Scholar 

  6. Dobrushin, R., Major, P.: Non-central limit theorems for non-linear functionals of Gaussian fields. Z. Wahrscheinlichkeitstheor. Verw. Geb.50, 27–52 (1979)

    Google Scholar 

  7. Feller, W.: An introduction to probability theory and its applications, vol. 2, 2nd ed. New York: Wiley 1971

    Google Scholar 

  8. Fox, R., Taqqu, M.: Noncentral limit theorems for quadratic forms in random variables having long-range dependence. Ann. Probab.13, 428–446 (1985)

    Google Scholar 

  9. Giraitis, L., Surgailis, D.: CLT and other limit theorems for functionals of Gaussian processes. Z. Wahrscheinlichkeitstheor. Verw. Geb.70, 191–212 (1985)

    Google Scholar 

  10. Itô, K.: Multiple Wiener integral. J. Math. Soc. Japan3, 157–169 (1951)

    Google Scholar 

  11. Itô, K.: Complex multiple Wiener integral. Jap. J. Math.22, 63–86 (1952)

    Google Scholar 

  12. Major, P.: Multiple Wiener-Itô integrals. Berlin: Springer 1981

    Google Scholar 

  13. Maruyama, G.: Nonlinear functionals of Gaussian stationary processes and their applications. In: Maruyama, G., Prokhorov J.V. (eds.) Proceedings, Third Japan-USSR symposium on probability theory. (Lect. Notes Math., vol. 550, pp. 375–378) Berlin Heidelberg New York: Springer 1976

    Google Scholar 

  14. Maruyama, G.: Wiener functionals and probability limit theorems I: the central limit theorems. Osaka J. Math.22, 697–732 (1985)

    Google Scholar 

  15. Rosenblatt, M.: Some limit theorems for partial sums of quadratic forms in stationary Gaussian variables. Z. Wahrscheinlichkeitstheor. Verw. Geb.49, 125–132 (1979)

    Google Scholar 

  16. Sun, T.C.: A central limit theorem for nonlinear functions of a normal stationary process. J. Math. Mech.12, 945–978 (1963)

    Google Scholar 

  17. Sun, T.C.: Some further results on central limit theorems for nonlinear functions of a normal stationary process. J. Math. Mech.14, 71–85 (1965)

    Google Scholar 

  18. Taqqu, M.: Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrscheinlichkeitstheor. Verw. Geb.31, 287–302 (1975)

    Google Scholar 

  19. Taqqu, M.: Convergence of iterated process of arbitrary Hermite rank. Z. Wahrscheinlichkeitstheor. Verw. Geb.50, 27–52 (1979)

    Google Scholar 

  20. Totoki, H.: Ergodic theory. Aarhus University Lecture Note Series, vol. 14 Aarhus University: Aarhus (1969)

    Google Scholar 

  21. Wiener, N.: The homogeneous chaos. Am. J. Math.60, 897–936 (1930)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Boston College, 02167, Chestnut Hill, MA, USA

    Daniel Chambers

  2. Department of Mathematics, University of Maryland, 20742, College Park, MD, USA

    Eric Slud

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  1. Daniel Chambers
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Chambers, D., Slud, E. Central limit theorems for nonlinear functionals of stationary Gaussian processes. Probab. Th. Rel. Fields 80, 323–346 (1989). https://doi.org/10.1007/BF01794427

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  • Received: 30 January 1985

  • Revised: 19 February 1987

  • Issue Date: September 1989

  • DOI: https://doi.org/10.1007/BF01794427

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Keywords

  • Hilbert Space
  • Stochastic Process
  • Probability Theory
  • Limit Theorem
  • Mathematical Biology
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