NON-CENTRAL LIMIT-THEOREMS FOR NONLINEAR FUNCTIONALS OF GAUSSIAN FIELDS

Let a stationary Gaussian sequence Xn, n=... -1,0,1, ... and a real function H(x) be given. We define the sequences {Mathematical expression}, n=... -1,0,1...; N=1,2, ... where ANare appropriate norming constants. We are interested in the limit behaviour as N→∞. The case when the correlation function r(n)=EX0Xn 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. © 1979 Springer-Verlag.