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

\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.