Geodesic
On sums of random variables with values in a Hilbert space
Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 4, pp. 760-763 
E. R. Vvedenskaya. On sums of random variables with values in a Hilbert space. Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 4, pp. 760-763. http://geodesic.mathdoc.fr/item/TVP_1983_28_4_a12/
@article{TVP_1983_28_4_a12,
     author = {E. R. Vvedenskaya},
     title = {On sums of random variables with values in {a~Hilbert} space},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {760--763},
     year = {1983},
     volume = {28},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1983_28_4_a12/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $H$ be a separable Hilbert space and $X1,X2,\dots$ be a sequence of independent random vectors identically and symmetrically distributed in $H$ such that $\mathbf P\{\|X_1\|>0\}>0$. Let $S_n=X_1+\dots+X_n$ and $$ \gamma_n(\alpha)=\inf\{R:\,\mathbf P\{\|S_n\|\le R\}\ge\alpha\},\qquad 0<\alpha<1. $$ We prove that if $\mathbf E\|X_1\|=\infty$ then $$ \mathbf P\{\limsup_{n\to\infty}\|S_n\|/\gamma_n(\alpha)=\infty\}=1. $$ In the finite-dimensional case the last equality is valid without any additional conditions as it follows from [4].