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

A variational formula on the Cram\'er function of series of independent random variables