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On an inequality of A. Ya. Khinchin. (Russian) Zbl 0568.60015
Problems of stability of stochastic models, Proc. Semin., Moskva 1984, 92-94 (1984).
[For the entire collection see Zbl 0554.00006.]
A classical inequality of Khinchin states that the \(L^ s\) norm \((1\leq s<\infty)\) of \(\sum^{n}_{k=1}c_ kr_ k\), where \(\{r_ k\}\) is the system of Rademacher functions, \(n\in N\) and \(c_ 1,c_ 2,...,c_ n\in R\), has a lower and upper bound of the form \(C_ s(\sum^{n}_{k=1}c^ 2_ k)^{1/2}\). The author generalizes this property for sequences of symmetric rv’s \(\{r_ k\}\) and for more general Banach space norms \((L^ s\) is replaced by a symmetric Banach space of measurable functions on [0,1] such that this Banach space is an interpolation between \(L^ 1\) and \(L^{\infty})\).
Reviewer: G.J.SzĂ©kely

60E15 Inequalities; stochastic orderings
60B05 Probability measures on topological spaces