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.
Type de document :
Pré-publication, Document de travail
2011
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https://hal.archives-ouvertes.fr/hal-00587560
Contributeur : Alexander Bulinski <>
Soumis le : mercredi 20 avril 2011 - 17:29:23
Dernière modification le : lundi 29 mai 2017 - 14:25:26
Document(s) archivé(s) le : jeudi 21 juillet 2011 - 03:11:46

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  • HAL Id : hal-00587560, version 1
  • ARXIV : 1104.4180

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UPMC | INSMI | PMA | USPC

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Alexander Bulinski. ON THE NEWMAN CONJECTURE. 2011. <hal-00587560>

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