On sums of random variables with values in a Hilbert space

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].