Exact Rosenthal-type bounds

Abstract

It is shown that, for any given $p\ge 5$, $A>0$ and $B>0$, the exact upper bound on $\mathsf{E}|\sum X_i{|}^p$ over all independent zero-mean random variables (r.v.'s) $X_1,\dots ,X_n$ 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.

Published at http://dx.doi.org/10.1214/14-AOP942 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Exact inequalities for sums of asymmetric random variables, with applications

Cited by in corpus (3)

• Sharp Rosenthal-type inequalities for mixtures and log-concave variables
• Positive-part moments via the characteristic functions, and more general expressions
• Maximal inequalities in quantum probability spaces