  Given a sequence $(X_n)$ of symmetrical random variables taking values in a
Hilbert space, an interesting open problem is to determine the conditions under
which the series $\sum_{n=1}^\infty X_n$ is almost surely convergent. For
independent random variables, it is well-known that if $\sum_{n=1}^\infty
\mathbb{E}(\|X_n\|^2) <\infty$, then $\sum_{n=1}^\infty X_n$ converges almost
surely. This has been extended to some cases of dependent variables (namely
negatively associated random variables) but in the general setting of dependent
variables, the problem remains open. This paper considers the case where each
variable $X_n$ is given as a linear combination $a_{n,1}Z_1+ \ldots
+a_{n,n}Z_n$ where $(Z_n)$ is a sequence of independent symmetrical random
variables of unit variance and $(a_{n,k})$ are constants. For Gaussian random
variables, this is the general setting. We obtain a sufficient condition for
the almost sure convergence of $\sum_{n=1}^\infty X_n$ which is also sufficient
for the almost sure convergence of $\sum_{n=1}^\infty \pm X_n$ for all
(non-random) changes of sign. The result is based on an important bound of the
mean of the random variable $\sup(\|X_1 + \ldots +X_k\|: 1\leq k \leq n)$ which
extends the classical L\'evy's inequality and has some independent interest.
