On the Newman Conjecture

TL;DR

This paper proves a modified version of Newman's conjecture, establishing a criterion for the central limit theorem to hold for a broad class of positively associated stationary random fields on multi-dimensional lattices.

Contribution

It extends the CLT validity criterion to higher-dimensional fields under a generalized covariance condition, using uniform integrability and variance representations.

Findings

The modified conjecture holds for all positive integers d.

A criterion for CLT validity is established for wider classes of fields.

The approach generalizes Lewis's theorem to multi-dimensional settings.

Abstract

We consider a random field, defined on an integer-valued d-dimensional lattice, with covariance function satisfying a condition more general than summability. Such condition appeared in the well-known Newman's conjecture concerning the central limit theorem (CLT) for stationary associated random fields. As was demonstrated by Herrndorf and Shashkin, the conjecture fails already for d=1. In the present paper, we show the validity of modified conjecture leaving intact the mentioned condition on covariance function. Thus we establish, for any positive integer d, a criterion of the CLT validity for the wider class of positively associated stationary fields. The uniform integrability for the squares of normalized partial sums, taken over growing parallelepipeds or cubes, plays the key role in deriving their asymptotic normality. So our result extends the Lewis theorem proved for sequences of random variables. A representation of variances of partial sums of a field using the slowly varying functions in several arguments is employed in essential way.