Ordinary and logarithmical convexity of moment generating function

TL;DR

This paper proves that the moment generating function (MGF) of centered random variables and vectors satisfying Kramer's condition exhibits both ordinary and logarithmic convexity, using Grand Lebesgue Spaces theory.

Contribution

It introduces new convexity properties of the MGF for certain random variables, leveraging the framework of Grand Lebesgue Spaces.

Findings

MGF is convex for centered variables under Kramer's condition

Logarithmic convexity of MGF is established

Uses Grand Lebesgue Spaces theory for proofs

Abstract

We establish an ordinary as well as a logarithmical convexity of the Moment Generating Function (MGF) for the centered random variable and vector (r.v.) satisfying the Kramer's condition. Our considerations are based on the theory of the so-called Grand Lebesgue Spaces.