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

The notion of the Hölder convolution is introduced. The main result is that, under general conditions on functions $L_1,\dots,L_n$, one has $${{(L_1\operatorname{\raisebox{.8pt}{\fbox{\tiny H}}}\cdots\operatorname{\raisebox{.8pt}{\fbox{\tiny H}}} L_n)}^*}^{-1}= {{L_1}^*}^{-1}+\dots+{{L_n}^*}^{-1},$$ where $\operatorname{\raisebox{.8pt}{\fbox{\tiny H}}}$ denotes the Hölder convolution and ${{L}^*}^{-1}$ is the function inverse to the Legendre-Fenchel transform $L^*$ of a given function $L$. General properties of the functions $L^*$ and ${{L}^*}^{-1}$ 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.