  In [11] it has been proved some variational formula on the Legendre-Fenchel
transform of the cumulant generating function (the Cram\'er function) of
Rademacher series with coefficients in the space $\ell^1$. In this paper we
show a generalization of this formula to series of a larger class of any
independent random variables with coefficients that belong to the space
$\ell^2$.
