The standard law of the iterated logarithm expresses that if $X_1, X_2, \ldots$ are iid real random variables with mean zero and variance $\sigma^2$, $$ \limsup_{n \to \infty} \frac {X_1 + \cdots + X_n}{\sqrt {2n \ln \ln n}} = \sigma $$ almost surely. Together with the same result for $-X_1, -X_2, \ldots$, the same limit holds true with absolute values around the sum $X_1 + \cdots + X_n$.

Is there a corresponding result when the $X_i$'s take values in a Hilbert space, and the absolute values are replaced by the Hilbert norm?