Coefficient Properties of Random Variable Sequences

Abstract

Burkholder [3] gave a simple necessary and sufficient condition, in terms of concentration functions, for independent sequences of random variables to have the Stein property. Here we find sufficient conditions for the Stein property without assuming that the random variable sequence is independent. Our conditions are also in terms of concentration functions, but in our case they are conditional concentration functions which specialize to those used by Burkholder. For some of our results, the sequence of random variables may be quite arbitrary; however, we usually assume it to be a martingale difference sequence satisfying certain regularity conditions.