Inequalities for Walsh like random variables (Q914233)

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.