(Quasi)additivity properties of the Legendre--Fenchel transform and its inverse, with applications in probability

TL;DR

This paper explores the additivity properties of the Legendre--Fenchel transform and its inverse, introducing the H"older convolution and applying these concepts to derive bounds on probability distributions.

Contribution

It establishes a general additivity property for the inverse Legendre--Fenchel transform of the H"older convolution, with applications to probability bounds.

Findings

Derived an upper bound on quantiles of sum of random variables.

Proved the inverse transform of the H"older convolution equals the sum of individual inverses.

Extended Legendre--Fenchel properties to probabilistic applications.

Abstract

The notion of the H\"older convolution is introduced. The main result is that, under general conditions on functions L_1, ..., L_n, the function inverse to the Legendre--Fenchel transform of the H\"older convolution of L_1, ..., L_n coincides with the sum of the inverses of the Legendre--Fenchel transforms of the individual functions L_1, ..., L_n. Applications to probability theory are presented. In particular, an upper bound on the quantiles of the distribution of the sum of random variables is given.