(Quasi)additivity Properties of the Legendre-Fenchel Transform and its Inverse, with Applications in Probability

I. Pinelis

Journal of convex analysis, Tome 24 (2017) no. 3, pp. 889-901

Résumé

\newcommand{\li}[1]{{{#1}^*}^{-1}} \newcommand{\fJt}{\operatorname{\raisebox{.8pt}{\fbox{\tiny H}}}} The notion of the H\"older convolution is introduced. The main result is that, under general conditions on functions $L_1,\dots,L_n$, one has $$ \li{(L_1\fJt\cdots\fJt L_n)}= \li{L_1}+\dots+\li{L_n}, $$ where $\fJt$ denotes the H\"older convolution and $\li L$ is the function inverse to the Legendre-Fenchel transform $L^*$ of a given function $L$. General properties of the functions $L^*$ and $\li L$ are discussed. Applications to probability theory are presented. In particular, an upper bound on the quantiles of the distribution of the sum of (possibly dependent) random variables is given.

Détail

Classification : 26A48, 26A51, 60E15
Mots-clés : Hoelder convolution, Legendre-Fenchel transform, probability inequalities, exponential inequalities, sums of random variables, exponential rate function, Cramer-Chernoff function, quantiles