(Quasi)additivity properties of the Legendre--Fenchel transform and its inverse, with applications in probability
Abstract
The notion of the H\"older convolution is introduced. The main result is that, under general conditions on functions L1, ..., Ln, the function inverse to the Legendre--Fenchel transform of the H\"older convolution of L1, ..., Ln coincides with the sum of the inverses of the Legendre--Fenchel transforms of the individual functions L1, ..., Ln. Applications to probability theory are presented. In particular, an upper bound on the quantiles of the distribution of the sum of random variables is given.
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