Let a stationary Gaussian sequence
X
n
,
n=... –1,0,1, ... and a real function H(x) be given. We define the sequences
YnN = \frac1AN ·åj = ( n - 1 )NnN - 1 H( Xj )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.
This paper contains results closely connected to those of the paper by Taqqu, Z. Wahrscheinlichkeitstheorie verw. Gebiete 50, 53–83 (1979). The investigations were done independently and at about the same time. Different methods were used
Dedicated to Professor Leopold Schmetterer on his sixtieth Birthday