WEAK LIMITS AND INTEGRALS OF GAUSSIAN COVARIANCES IN
BANACH SPACES
J. M. A. M. van Neerven
L. Weis
Abstract: Let 
be a separable real Banach space not containing an isomorphic copy of 
Let 
be a subset of 
with the property that each 
is the covariance of the
centred Gaussian measure 
on 
We show that the weak operator closure of 
consists of Gaussian covariances again, provided that
If in
addition
has type 2, the same conclusion holds for the weak operator closure of the
convex hull of
As an application, sufficient conditions are obtained for the integral of
Gaussian covariance operators to be a Gaussian covariance. Analogues of these results are
given for the class of
-radonifying operators from a separable real Hilbert space
into
2000 AMS Mathematics Subject Classification: Primary 28C20; Secondary: 35R15,
60B11, 60H05.
Key words and phrases: Gaussian Radon measure, covariance operator, 
-radonifying
operator, Fatou lemma, type 2, cotype 2, weak operator topology.
