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

TL;DR

This paper generalizes a variational formula for the Cramér function from Rademacher series to a broader class of independent random variable series with coefficients in ^2, expanding its applicability.

Contribution

It extends the variational formula for the Crame9r function from Rademacher series to series of arbitrary independent random variables with ^2 coefficients.

Findings

Generalized the variational formula to a wider class of random variables.

Provided a new theoretical framework for analyzing large deviations.

Expanded the applicability of the Crame9r function analysis.

Abstract

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 $\ell^2$.