Geodesic
On sums of random vectors with values in a~Hilbert space
Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 2, pp. 354-358

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Let $H$ be a separable Hilbert space and $X_1,X_2,\dots$ be a sequence of independent random vectors with values in $H$ and with a common symmetric probability distribution $R$. Let $S_n=X_1+X_2+\dots+X_n$. We prove that there exists $R$ such that for some $b_n>0$ $$ \|S_n|^2b_n^{-1}\to 1\qquad\text{in probability.} $$ There exist no such $R$ in linite-dimensional case, but in general infinite-dimensional case $\|S_n\|^2b_n^{-1}$ may converge to 1 with probability 1. 
@article{TVP_1983_28_2_a8,
     author = {Yu. V. Prohorov},
     title = {On sums of random vectors with values in {a~Hilbert} space},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {354--358},
     publisher = {mathdoc},
     volume = {28},
     number = {2},
     year = {1983},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1983_28_2_a8/}
}
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Yu. V. Prohorov. On sums of random vectors with values in a~Hilbert space. Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 2, pp. 354-358. http://geodesic.mathdoc.fr/item/TVP_1983_28_2_a8/