Exact rosenthal-type bounds

Document Type

Article

Publication Date

1-1-2015

Abstract

© Institute of Mathematical Statistics, 2015. It is shown that, for any given p ≥ 5, A> 0 and B> 0, the exact upper bound on E|ΣXi|p over all independent zero-mean random variables (r.v.'s) X1, . . . , Xn such that ΣEX2i = B and Σ E|Xi|p = A equals cpE|πλ -λ|p, where (λ, c) ε (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.

Publication Title

Annals of Probability

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