Exact Rosenthal-type bounds

TL;DR

This paper establishes exact upper bounds for the p-th moments of sums of independent zero-mean random variables with fixed second and p-th moments, using Poisson distributions and variational calculus of infinitely divisible distributions.

Contribution

It provides the first exact Rosenthal-type bounds for moments of sums of independent variables with specified moments, employing a novel variational approach.

Findings

Derived explicit formulas for upper bounds involving Poisson distributions.

Developed a calculus of variations for moments of infinitely divisible distributions.

Extended results to more general settings and related inequalities.

Abstract

It is shown that, for any given $p\ge5$, $A>0$ and $B>0$, the exact upper bound on $\mathsf{E}|\sum X_i|^p$ over all independent zero-mean random variables (r.v.'s) $X_1,\ldots,X_n$ such that $\sum\mathsf{E}X_i^2=B$ and $\sum\mathsf{E}|X_i|^p=A$ equals $c^p\mathsf{E}|\Pi_{\lambda}-\lambda|^p$, where $(\lambda ,c)\in(0,\infty)^2$ is the unique solution to the system of equations $c^p\lambda=A$ and $c^2\lambda=B$, and $\Pi_{\lambda}$ is a Poisson r.v. with mean $\lambda$. In fact, a more general result is obtained, as well as other related ones. As a tool used in the proof, a calculus of variations of moments of infinitely divisible distributions with respect to variations of the L\'{e}vy characteristics is developed.