Central Limit Theorem for Positively Associated Stationary Random Fields

Positively associated stationary random fields on d-dimensional integral lattice arise in various models of mathematical statistics, percolation theory, statistical physics, and reliability theory. In this paper, we shall be concerned with a field with covariance functions satisfying a more general condition than summability. A criterion for the validity of the central limit theorem (CLT) for partial sums of a field from this class is established. The sums are taken over an increasing nest of parallelepipeds or cubes. The well-known conjecture of Newman stated that for an associated stationary random field the above condition on the covariance function should force the CLT to hold. As was shown by N. Herrndorf and A. P. Shashkin, this conjecture fails already for d = 1. In the present paper, the uniform integrability of the squared partial sums is shown as being of key importance for the CLT to hold. Thus, an extension of Lewis's theorem proved for a sequence of random variables is obtained. Also, it is indicated how to modify Newman's conjecture for any d. A representation of variances of partial sums of a field by means of slowly varying functions of several arguments is used in an essential way.