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Best constants in moment inequalities for linear combinations of independent and exchangeable random variables. (English) Zbl 0564.60020

Generalizing Khintchine’s inequality, H. P. Rosenthal [Isr. J. Math. 8, 273-303 (1970; Zbl 0213.193)] proved that for \(2<p<\infty\) there exists a constant \(B_ p\) with the following property: If \(X_ 1,X_ 2,..\). are independent symmetric random variables in \(L_ p\), \(S_ n=\sum^{n}_{i=1}X_ i,\) and \(M=Max\{\| S_ n\|_ 2,(\sum^{n}_{i=1}\| X_ i\|^ p_ p)^{1/p}\},\) then \(M\leq \| S_ n\|_ p\leq B_ pM\). Previous proofs yielded only exponential growth rates for \(B_ p\) for \(p\to \infty.\)
Here, it is shown that the growth rate is p/log p, while the growth rate in Khintchine’s inequality is \(\sqrt{p}\). The authors also study the growth rates of other related inequalities.
Reviewer: U.Krengel

MSC:

60E15 Inequalities; stochastic orderings
60G50 Sums of independent random variables; random walks
60G42 Martingales with discrete parameter

Citations:

Zbl 0213.193
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