Journal of Theoretical Probability

, Volume 4, Issue 2, pp 407–415 | Cite as

A mixture-type limit theorem for nonlinear functions of Gaussian sequences

  • Hwai-Chung Ho
  • Tze-Chien Sun
Article

Abstract

We show that, for a certain class of nonlinear functions of Gaussian sequences, the limiting distribution of normalized sums of the nonlinear function values of a sequence is the convolution of a Gaussian distribution with another non-Gaussian distribution.

Key Words

nonlinear functions of Gaussian sequences Gaussian distribution convolution 

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References

  1. 1.
    Breuer, P., and Major, P. (1983). Central limit theorems for non-linear functionals of Gaussian fields.J. Mult. Analysis 13, 425–441.Google Scholar
  2. 2.
    Chambers, D., and Slud, E. (1989). Central limit theorems for non-linear functionals of stationary Gaussian processes.Prob. Theor. Relative Fields 80, 323–346.Google Scholar
  3. 3.
    Dobrushin, R. L., and Major, P. (1979). Non-central limit theorems for non-linear functionals of Gaussian fields.Z. Wahrsch. verw. Gebiete 50, 27–52.Google Scholar
  4. 4.
    Ho, H. C., and Sun, T. C. (1987). A central limit theorem for non-instantaneous filters of a stationary Gaussian process.J. Mult. Analysis 22, 144–155.Google Scholar
  5. 5.
    Ho, H. C., and Sun, T. C. (1990). Limiting distributions of non-linear vector functions of stationary Gaussian processes.Ann. Prob. 18, 1159–1173.Google Scholar
  6. 6.
    Major, P. (1981).Multiple Wiener-Ito Integrals. Lecture Notes in Mathematics, Vol. 849, Springer-Verlag, New York-Berlin.Google Scholar
  7. 7.
    Major, P. (1981). Limit theorems for non-linear functionals of Gaussian sequences.Z. Wahrsch. verw. Gebiete 57, 129–158.Google Scholar
  8. 8.
    Rosenblatt, M. (1987). Remarks on limit theorems for non-linear functionals of Gaussian sequences.Prob. Theor. Fields 75, 1–10.Google Scholar
  9. 9.
    Taqqu, M. (1979). Convergence of integrated processes of arbitrary Hermite rank.Z. Wahrsch. verw. Gebiete 50, 53–83.Google Scholar
  10. 10.
    Zygmund, A. (1959).Trigonometric Series, Vols. 1 and 2, 2nd ed. Oxford University Press, London and New York.Google Scholar

Copyright information

© Plenum Publishing Corporation 1991

Authors and Affiliations

  • Hwai-Chung Ho
    • 1
    • 3
  • Tze-Chien Sun
    • 2
  1. 1.Institute of Statistical ScienceAcademia SinicaTaipeiTaiwan
  2. 2.Department of MathematicsWayne State UniversityDetroit
  3. 3.National Sun Yat-Sen UniversityKaohsiungTaiwan

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