Vol. 17, No. 1, 1966

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Gaussian measures in function space

Lawrence A. Shepp

Vol. 17 (1966), No. 1, 167–173

Two Gaussian measures are either mutually singular or equivalent. This dichotomy was first discovered by Feldman and Hajek (independently). We give a simple, almost formal, proof of this result, based on the study of a certain pair of functionals of the two measures. In addition we show that two Gaussian measures with zero means and smooth Polya-type covariances (on an interval) are equivalent if and only if the right-hand slopes of the covariances at zero are equal.

Mathematical Subject Classification
Primary: 60.40
Received: 6 October 1964
Revised: 20 November 1964
Published: 1 April 1966
Lawrence A. Shepp