Polarized random variables: Maximal correlations and common information

Abstract

New theorems are established regarding polarized Bernoulli random variables: (i) The maximal correlations between polarized Bernoulli variables converge to zero or one as do the conditional entropy and Bhattacharyya parameters; (ii) The graphical model of polarized Bernoulli variables provides a way to compute pair-wise and higher-order correlations; (iii) The Wyner common information between two sequences of correlated random variables may be extracted using Arikan's polar transform which leads to a low-complexity solution to the Wyner network. In addition, a joint polarization theorem is provided involving common information.