Structural theorems for multiplicative systems of functions
Let us define several key concepts: A sequence \((X_ i)\) of random variables is called a multiplicative system (MS system), if E \(X_{i_ 1}X_{i_ 2}...X_{i_ n}=0\) for every sequence of different indices. If moreover E \(X^ 2_{i_ 0}X_{i_ 1}...X_{i_ n}=0\) for every sequence of different indices, \((X_ i)\) is called a strongly multiplicative system (SMS system). If E \(X_{i_ 1}X_{i_ 2}...X_{i_ m}=0\) for every \(m\leq k\) and every sequence of indices \(i_ 1<...<i_ m\) then the sequence \((X_ i)\) is called an MS system of order k. Let X and Y be random variables with values in a Banach space F. We say that X is strongly dominated by Y if E f(X)\(\leq E f(Y)\) for every continuous, convex and nonnegative function \(f: F\to R^+\), provided both integrals exist. The main result is as follows: Theorem 1. Let \(X_ 1,...,X_ n\) be an MS system of order \(k\leq n\) and \(Y_ 1,...,Y_ n\) be an SMS system. If \(0<D^ 2Y_ i<\infty\) and \(\sup | X_ iY_ i| \leq (k/n)D^ 2Y_ i\) for \(i=1,2,...,n\), then \((X_ 1,...,X_ n)\) is strongly dominated by \((Y_ 1,...,Y_ n).\) The above structural theorem explains why multiplicative systems have a structure similar to the structure of Rademacher system; in particular it explains similar properties of lacunary trigonometric systems and the Rademacher system. The paper also shows that the Rademacher system dominates a uniformly bounded p-weakly multiplicative system in some sense. As applications of the results, the paper discusses convergence almost surely, moment inequalities, strong laws of large numbers and laws of the iterated logarithm.
scientific article; zbMATH DE number 4015790
scientific article