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

Abstract

The notion of the Hölder 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 Hölder 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.

Version 2: three references added. Version 3: The main result is strengthened from an inequality to an identity. Propositions 1.2, 1.4, and 1.5 are added, as well as four new references to literature. The title and abstract have changed