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WROCŁAW UNIVERSITY
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Contents of PMS, Vol. 6, Fasc. 2,
pages 151 - 159
 

STATISTICAL CHARACTERIZATIONS OF GAUSSIAN MEASURES ON A HILBERT SPACE

Harald Luschgy

Abstract: Let X ,...,X 1 n be i.i.d. random vectors with values in a real separable Hilbert space. We consider the problem of estimating the mean of X 1 under quadratic loss and discuss analogues of characteristic properties of normally distributed real random variables. It is shown that there exists an equivariant sufficient linear statistic iff X 1 is Gaussian. Further the optimality of the sample mean X in the class of all equivariant or unbiased estimators is a characteristic property of Gaussian random vectors.

2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;

Key words and phrases: -

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