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

  • R. L. Dobrushin
  • P. Major
Article

Summary

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.

Keywords

Correlation Function Stochastic Process Probability Theory Limit Theorem Mathematical Biology 
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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

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