The Annals of Probability

Exact Rosenthal-type bounds

Iosif Pinelis

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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.

Article information

Ann. Probab., Volume 43, Number 5 (2015), 2511-2544.

Received: June 2013
Revised: May 2014
First available in Project Euclid: 9 September 2015

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Zentralblatt MATH identifier

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60E07: Infinitely divisible distributions; stable distributions

Rosenthal inequality bounds on moments sums of independent random variables probability inequalities calculus of variations infinitely divisible distributions Lévy characteristics


Pinelis, Iosif. Exact Rosenthal-type bounds. Ann. Probab. 43 (2015), no. 5, 2511--2544. doi:10.1214/14-AOP942.

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