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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

##### MSC:
 60E15 Inequalities; stochastic orderings 60B99 Probability theory on algebraic and topological structures
##### Keywords:
Banach space valued variables; symmetric distribution