(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

, 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}, ( L 1 H ⁡ ⋯ H ⁡ L n ) ∗ − 1 = L 1 ∗ − 1 + ⋯ + L n ∗ − 1 ,

where

\operatorname{\raisebox{.8pt}{\fbox{\tiny H}}} H ⁡

denotes the Hölder convolution and

{{L}^*}^{-1} L ∗ − 1

is the function inverse to the Legendre-Fenchel transform

L^* L ∗

of a given function

L L

. General properties of the functions

L^* L ∗

and

{{L}^*}^{-1} 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.