Geodesic
A Characterization of Normal Distributions in Hilbert Space
Teoriâ veroâtnostej i ee primeneniâ, Tome 2 (1957) no. 4, pp. 475-477

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Theorem 1 gives an extension of the well known result of [2] to the case of random elements in Hilbert space. 
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     title = {A {Characterization} of {Normal} {Distributions} in {Hilbert} {Space}},
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Yu. V. Prokhorov; M. Fish. A Characterization of Normal Distributions in Hilbert Space. Teoriâ veroâtnostej i ee primeneniâ, Tome 2 (1957) no. 4, pp. 475-477. http://geodesic.mathdoc.fr/item/TVP_1957_2_4_a5/