
Journal of Convex Analysis 24 (2017), No. 3, 889901 Copyright Heldermann Verlag 2017 (Quasi)additivity Properties of the LegendreFenchel Transform and its Inverse, with Applications in Probability Iosif Pinelis Dept. of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, U.S.A. ipinelis@mtu.edu [Abstractpdf] \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 LegendreFenchel 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, LegendreFenchel transform, probability inequalities, exponential inequalities, sums of random variables, exponential rate function, CramerChernoff function, quantiles. MSC: 26A48, 26A51; 60E15 [ Fulltextpdf (150 KB)] for subscribers only. 