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Short Communications

On admissible translations of weighted Gaussian measures

G. N. Sytaya

Kiev

Abstract: A family of Gaussian measures $\{\mu_{\Sigma\alpha_iA_i}(dx)\}$ in a Hilbert space $H$ with characteristic functional $\exp\Bigr\{-\frac12\sum_{i=1}^m\alpha_i(A_iz,z)\Bigr\}$ and their weighted sums (1) is considered. The set of admissible translations of (1) (as well as each $\mu_{\Sigma\alpha_iA_i}$) is shown to coincide with $\Bigl(\sum_{i=1}^mA_i\Bigr)^{1/2}H$. For $m=1$ an expression for the corresponding density is found.

Received: 24.11.1967

 English version: Theory of Probability and its Applications, 1969, 14:3, 506–509

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