Covariance linear functionals on doubly stochastic measures

Doubly stochastic measures are joint distributions with uniform marginals. They are intimately tied to doubly stochastic operators, Markov processes, and dynamical systems. They form a complicated convex set which can be studied through its extreme points. A convariance linear functional is a statistic which can be used in this study to identify extreme points and to yield other information regarding the connection to the operator, process and dynamical system. In turn, the covariance gives rise to a covariance matrix which can be used in a study of the convergence of the iterates of the operator.