Geodesic
On admissible translations of weighted Gaussian measures
Teoriâ veroâtnostej i ee primeneniâ, Tome 14 (1969) no. 3, pp. 527-531

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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. 
@article{TVP_1969_14_3_a13,
     author = {G. N. Sytaya},
     title = {On admissible translations of weighted {Gaussian} measures},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {527--531},
     publisher = {mathdoc},
     volume = {14},
     number = {3},
     year = {1969},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1969_14_3_a13/}
}
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G. N. Sytaya. On admissible translations of weighted Gaussian measures. Teoriâ veroâtnostej i ee primeneniâ, Tome 14 (1969) no. 3, pp. 527-531. http://geodesic.mathdoc.fr/item/TVP_1969_14_3_a13/