Tail bounds on random series in Hilbert space

Let $X_n$, $n \in \mathbb {N}$, be independent $\pm 1$ symmetric random variables, and $a_n$, $n \in \mathbb {N}$, be a sequence in a Hilbert space $H$ such that $\sum_n \|a_n\|^2 < \infty$, say $= 1$. Set $Z = \| \sum_n a_n X_n \|$ so that $E(Z^2) = \sum_n \|a_n\|^2 = 1$. Since $E(Z) \leq 1$, it is of interest to bound the probability $ P( Z > 1 + t)$, $t >0$. What would be a (sharp, exponential?) tail estimate in terms of the coefficients $a_n$, and in which range of $t >0$?