Non-central limit theorems for non-linear functional of Gaussian fields

  • R. L. Dobrushin
  • P. Major


Let a stationary Gaussian sequence X n , n=... −1,0,1, ... and a real function H(x) be given. We define the sequences \(Y_n^N = \frac{1}{{A_N }} \cdot \sum\limits_{j = \left( {n - 1} \right)N}^{nN - 1} {H\left( {X_j } \right)}\),n=... −1,0,1...; N=1,2, ... where A N are appropriate norming constants. We are interested in the limit behaviour as N→∞. The case when the correlation function r(n)=EX 0 X n tends slowly to 0 is investigated. In this situation the norming constants A> N tend to infinity more rapidly than the usual norming sequence A> N =√N. Also the limit may be a non-Gaussian process. The results are generalized to the case when the parameter-set is multi-dimensional.


Correlation Function Stochastic Process Probability Theory Limit Theorem Mathematical Biology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Dobrushin, R.L.: Gaussian and their subordinated self-similar random fields. Ann. Probability 7. No. 1, 1–28 (1979)Google Scholar
  2. 2.
    Dobrushin, R.L.: Automodel generalized random fields and their renorm group. In volume: Multi-component stochastic systems (in Russian) p. 179–213. Moscow: Nauka, 1978. English edition: New York: Marcel Dekker Ed. [in preparation]Google Scholar
  3. 3.
    Dobrushin, R.L., Minlos, R.A.: Polynomials of random functions. (in Russian) Achievements, Uspeschi, Math. Sci. XXXII. No 2 194, 67–122 (1977)Google Scholar
  4. 4.
    Dobrushin, R.L., Takahashi, J.: Self-similar Gaussian fields. [To appear]Google Scholar
  5. 5.
    Ibragimov, I.A., Linnik, J.V.: Independent and stationary sequences of random variables. Groningen: Walters-Noordhoff 1971Google Scholar
  6. 6.
    Rosenblatt, M.,: Independence and dependence. Proc. 4th Sympos. Math. Statist. Probability pp. 411–443. Univ. California: Berkeley University Press 1961Google Scholar
  7. 7.
    Simon, B.: The P(φ) 2 Euclidean (Quantum) field theory. Princeton: Princeton University Press 1974Google Scholar
  8. 8.
    Taqqu, M.S.: Weak convergence to Fractional Brownian Motion and the Rosenblatt Process. Z. Wahrscheinlichkeitstheorie verw. Gebiete 31, 287–302 (1975)Google Scholar
  9. 9.
    Zygmund, A.: Trigonometric series. Cambridge: Cambridge University Press 1959Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • R. L. Dobrushin
    • 1
  • P. Major
    • 2
  1. 1.Institute for Problems of Information TransmissionMoscow
  2. 2.Mathematical Institute of the Hungarian Academy of SciencesBudapestHungary

Personalised recommendations