On sums of random vectors with values in a Hilbert space

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.