Asymptotic Behavior of the Autocovariance Function and Violation of Strong Mixing

Abstract

We discuss violation of the strong mixing condition for a class of multidimensional stochastic processes which includes cases of physical interest like ferromagnetic systems. We give a sufficient condition for strong mixing violation in terms of the asymptotic behavior of the autocorrelation function over large distances. As an important example to which our criterion applies,we discuss the two dimensional Ising model at the critical point. Connections with other problems in statistical physics and probability theory are briefly reviewed.