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
