  Let $X_1,X_2,\ldots,X_n$ be independent random variables and
$S_k=\sum_{i=1}^k X_i$. We show that for any constants $a_k$, \[
\Pr(\max_{1\leq k\leq n}||S_{k}|-a_{k}|>11t)\leq 30 \max_{1\leq k\leq
n}\Pr(||S_{k}|-a_{k}|>t). \] We also discuss similar inequalities for sums of
Hilbert and Banach space valued random vectors.
