Skitovich–Darmois Theorem for Discrete and Compact Totally Disconnected Abelian Groups

Abstract

The classic Skitovich–Darmois theorem states that the Gaussian distribution on the real line can be characterized by the independence of two linear forms of n independent random variables. We generalize the Skitovich–Darmois theorem to discrete Abelian groups, compact totally disconnected Abelian groups, and some other classes of locally compact Abelian groups. Unlike the previous investigations, we consider n linear forms of n independent random variables.