Geodesic
Limit theorems for sums of independent random variables with values in Hilbert space
Teoriâ veroâtnostej i ee primeneniâ, Tome 17 (1972) no. 2, pp. 209-227

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Necessary and sufficient conditions are found for the distributions of sums (in the series scheme) of independent Hilbert space valued variables to converge to an arbitrary fixed distribution. 
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     author = {V. M. Kruglov},
     title = {Limit theorems for sums of independent random variables with values in {Hilbert} space},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {209--227},
     publisher = {mathdoc},
     volume = {17},
     number = {2},
     year = {1972},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1972_17_2_a0/}
}
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V. M. Kruglov. Limit theorems for sums of independent random variables with values in Hilbert space. Teoriâ veroâtnostej i ee primeneniâ, Tome 17 (1972) no. 2, pp. 209-227. http://geodesic.mathdoc.fr/item/TVP_1972_17_2_a0/