ON THE NEWMAN CONJECTURE - Archive ouverte HAL Access content directly
Preprints, Working Papers, ... Year : 2011

ON THE NEWMAN CONJECTURE

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.
Fichier principal
Vignette du fichier
bulinski_CLT_LPMA_UPMC.pdf (145.16 Ko) Télécharger le fichier
Origin : Files produced by the author(s)
Loading...

Dates and versions

hal-00587560 , version 1 (20-04-2011)

Identifiers

Cite

Alexander Bulinski. ON THE NEWMAN CONJECTURE. 2011. ⟨hal-00587560⟩
401 View
49 Download

Altmetric

Share

Gmail Facebook X LinkedIn More