Exact Rosenthal-type bounds

Pinelis, Iosif ^[1]
1. [1] Michigan Technological University
  
  Michigan Technological University
  
  City of Houghton, Estados Unidos
Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 43, Nº. 5, 2015, págs. 2511-2544
Idioma: inglés
DOI: 10.1214/14-AOP942
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Resumen
- It is shown that, for any given p≥5, A>0 and B>0, the exact upper bound on E|∑Xi|p over all independent zero-mean random variables (r.v.’s) X1,…,Xn such that ∑EX2i=B and ∑E|Xi|p=A equals cpE|Πλ−λ|p, where (λ,c)∈(0,∞)2 is the unique solution to the system of equations cpλ=A and c2λ=B, and Πλ is a Poisson r.v. with mean λ. 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.