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Extremal problems in moment inequalities. (Russian) Zbl 0579.60018
Predel’nye Teoremy Teorii Veroyatnostej, Tr. Inst. Mat. 5, 56-75 (1985).
Let \(X_ 1,X_ 2,..\). be independent random variables with zero expectation. Then, for a function \(\phi\) satisfying some kind of convexity and differentiability assumptions, it is shown that \(E[\phi (\sum^{n}_{i=1}X_ i+x)]\geq \sum^{n}_{i=1}E[\phi (X_ i+x)]- (n\quad -1)\phi (x).\) Here the \(X_ i\) may have values in a separable Banach space. The assumptions on \(\phi\) are such that \(\phi (x)=\| x\|^ t\), \(t\geq 8\), is a special case, thus it covers earlier inequalities known for symmetric distributions. While, for symmetric distributions, the corresponding inequality is valid for \(t\geq 2\), it is not true in general. A simple example is included to show that such inequalities might fail for \(2<t<3\). For specific \(\phi\) ’s, optimal bounds are obtained for the left hand side above.
Reviewer: J.Galambos

60E15 Inequalities; stochastic orderings
60B99 Probability theory on algebraic and topological structures