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Supremum of an even moment of sums of independent random variables. (English. Russian original) Zbl 0731.60014

Sib. Math. J. 32, No. 1, 139-141 (1991); translation from Sib. Mat. Zh. 32, No. 1(185), 171-173 (1991).
The main result gives the following Theorem: For every integer \(m>1\), \[ \sup E(X_ 1+...+X_ n)^{2m}=\phi^{2m}E(\eta_{\lambda}- \lambda)^{2m}, \] where the supremum is taken over all \(n\geq 1\) and all independent random variables \(X_ 1,X_ 2,...,X_ n\) such that E \(X_ i=0\), \(1\leq i\leq n\), \(E X^ 2_ 1+...+E X^ 2_ n=B,\) and \(E X_ 1^{2m}+...+E X_ n^{2m}=A;\phi =(A/B)^{1/(2m-2)}\), and the random variable \(\eta_{\lambda}\) has Poisson distribution with \(\lambda =(B^ m/A)^{1/(m-1)}\).
Reviewer: Z.Rychlik (Lublin)

MSC:

60E05 Probability distributions: general theory
60G50 Sums of independent random variables; random walks
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