Abstract

The notion of the Hölder convolution is introduced. The main result is that, under general conditions on functions L1,…,LnL_1,\dots,L_n, one has (L1H⁡⋯H⁡Ln)∗−1=L1∗−1+⋯+Ln∗−1,{{(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 H⁡\operatorname{\raisebox{.8pt}{\fbox{\tiny 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 LL. 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.

Author information

Contact details are reproduced from the original publication and may be historical.

Iosif Pinelis

Dept. of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, U.S.A.

ipinelis@mtu.edu

Keywords

  • Hölder convolution
  • Legendre-Fenchel transform
  • probability inequalities
  • exponential inequalities
  • sums of random variables
  • exponential rate function
  • Cramér-Chernoff function
  • quantiles

Mathematics Subject Classification

26A48, 26A51; 60E15

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Suggested citation

I. Pinelis. “(Quasi)additivity Properties of the Legendre-Fenchel Transform and its Inverse, with Applications in Probability.” Journal of Convex Analysis 24 (2017), No. 3, 889–901. https://doi.org/10.68381/jca24052

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Published by Heldermann Verlag, 2017. Rights now held by Banach Press LLC.