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

Abstract

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.