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Journal of Convex Analysis 24 (2017), No. 3, 889--901
Copyright Heldermann Verlag 2017

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

Iosif Pinelis
Dept. of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, U.S.A.


\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.

Keywords: Hoelder convolution, Legendre-Fenchel transform, probability inequalities, exponential inequalities, sums of random variables, exponential rate function, Cramer-Chernoff function, quantiles.

MSC: 26A48, 26A51; 60E15

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