Supremum of an even moment of sums of independent random variables (Q808084)

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 \(\lambda =(B^ m/A)^{1/(m-1)}\).