Probability Theory and Related Fields

, Volume 80, Issue 3, pp 323–346 | Cite as

Central limit theorems for nonlinear functionals of stationary Gaussian processes

  • Daniel Chambers
  • Eric Slud
Article

Summary

Let X=(Xt,t∈ℝ) be a stationary Gaussian process on (Ω, ℱ,P), letH(X) be the Hilbert space of variables inL2 (Ω,P) which are measurable with respect toX, and let (Us,s∈ℝ) be the associated family of time-shift operators. We sayYH(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).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Billingsley, P.: Convergence of probability measures. New York: Wiley 1968Google Scholar
  2. 2.
    Billingsley, P.: Probability and measure. New York: Wiley 1979Google Scholar
  3. 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. 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. 5.
    Dobrushin, R.: Gaussian and their subordinated self-similar random generalized fields. Ann. Probab.7, 1–28 (1979)Google Scholar
  6. 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. 7.
    Feller, W.: An introduction to probability theory and its applications, vol. 2, 2nd ed. New York: Wiley 1971Google Scholar
  8. 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. 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. 10.
    Itô, K.: Multiple Wiener integral. J. Math. Soc. Japan3, 157–169 (1951)Google Scholar
  11. 11.
    Itô, K.: Complex multiple Wiener integral. Jap. J. Math.22, 63–86 (1952)Google Scholar
  12. 12.
    Major, P.: Multiple Wiener-Itô integrals. Berlin: Springer 1981Google Scholar
  13. 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 1976Google Scholar
  14. 14.
    Maruyama, G.: Wiener functionals and probability limit theorems I: the central limit theorems. Osaka J. Math.22, 697–732 (1985)Google Scholar
  15. 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. 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. 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. 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. 19.
    Taqqu, M.: Convergence of iterated process of arbitrary Hermite rank. Z. Wahrscheinlichkeitstheor. Verw. Geb.50, 27–52 (1979)Google Scholar
  20. 20.
    Totoki, H.: Ergodic theory. Aarhus University Lecture Note Series, vol. 14 Aarhus University: Aarhus (1969)Google Scholar
  21. 21.
    Wiener, N.: The homogeneous chaos. Am. J. Math.60, 897–936 (1930)Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • Daniel Chambers
    • 1
  • Eric Slud
    • 2
  1. 1.Department of MathematicsBoston CollegeChestnut HillUSA
  2. 2.Department of MathematicsUniversity of MarylandCollege ParkUSA

Personalised recommendations