A characterization of a gaussian process in terms of sufficient estimators

In this paper we consider the estimation problem in a continuous time linear model. We establish that, under certain covariance structure of the process, if the best linear unbiased estimator for the expectation of the process is sufficient then the process involved has a Gaussian distribution. In particular, this implies that, under some conditions, the linear sufficiency and ordinary sufficiency properties are equivalent if and only if the distribution of the process is Gaussian.