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September 2015 Exact Rosenthal-type bounds
Iosif Pinelis
Ann. Probab. 43(5): 2511-2544 (September 2015). DOI: 10.1214/14-AOP942


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.


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Iosif Pinelis. "Exact Rosenthal-type bounds." Ann. Probab. 43 (5) 2511 - 2544, September 2015.


Received: 1 June 2013; Revised: 1 May 2014; Published: September 2015
First available in Project Euclid: 9 September 2015

zbMATH: 1336.60033
MathSciNet: MR3395468
Digital Object Identifier: 10.1214/14-AOP942

Primary: 60E15
Secondary: 60E07

Keywords: bounds on moments , calculus of variations , Infinitely divisible distributions , Lévy characteristics , Probability inequalities , Rosenthal inequality , Sums of independent random variables

Rights: Copyright © 2015 Institute of Mathematical Statistics


Vol.43 • No. 5 • September 2015
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