Exact Rosenthal-type bounds

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 Lévy characteristics is developed.