Sets of Uniqueness of Series of Stochastically Independent Functions

Abstract

It is shown that, for every sequence (fn) of stochastically independent functions defined on [0, 1]-of mean zero and variance one, uniformly bounded by M -if the series ∞ n=1 anfn converges to some constant on a set of positive measure, then there are only finitely many non-null coefficients an, extending similar results by Stechkin and Ul'yanov on the Rademacher system. The best constant C M is computed such that for every such sequence (fn) any set of measure strictly less than C M is a set of uniqueness for (fn).

Co-countable sets of uniqueness for series of independent random variables

“…For the Rademacher system, every set of measure strictly greater than 1/2 determines, while for any set of positive measure it can be proved that any series which converges to zero on it is actually a finite sum [7]. This result was extended to some systems of independent random variables in [4].…”