## 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

### Keywords:

exponential growth rates; Khintchine’s inequality

Zbl 0213.193
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