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Inequalities for Walsh like random variables. (English) Zbl 0701.60014

Let \(\{X_ n\}^{\infty}_{n=1}\) be a sequence of mean-zero independent random variables on an arbitrary probability space, and assume that \(0<\delta \leq | X_ n| \leq K\) for all n, where \(\delta\) and K are fixed constants. For \(1<p<\infty\) and m a positive integer, define \[ C(p,m)=(16p/\log p)((5p^ 2\sqrt{2})/(p-1))^{m- 1}(K/\delta)^ m. \] The author establishes the following inequalities for any function f in the linear span of the set of products of m or fewer of the \(X_ n:\)
\(\| f\|_ p\leq C(p,m)\| f\|_ 2\) (when \(2<p<\infty);\)
\(\| f\|_ 2\leq C(q,m)\| f\|_ p\) (when \(1<p<2\) and \(1/p+1/q=1);\)
\(\| f\|_ 2\leq C(4,m)^ 2\| f\|_ 1.\)
Since the function \(p\mapsto \| f\|_ p\) is nondecreasing in p, it follows that the p-th mean of such functions f is norm-equivalent to the second moment.
Reviewer: B.K.Horkelheimer

MSC:

60E15 Inequalities; stochastic orderings
60A99 Foundations of probability theory
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