On the Newman Conjecture

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

Pl\"unnecke inequalities for measure graphs with applications

We generalize Petridis's new proof of Pl\"unnecke's graph inequality to graphs whose vertex set is a measure space. Consequently, this gives new Pl\"unnecke inequalities for measure preserving actions which enable us to deduce, via a Furstenberg correspondence principle, Banach density estimates in countable abelian groups that improve on those given by Jin.Comment: 24 pages, 1 figur

Strong invariance principle for dependent random fields

A strong invariance principle is established for random fields which satisfy dependence conditions more general than positive or negative association. We use the approach of Cs\"{o}rg\H{o} and R\'{e}v\'{e}sz applied recently by Balan to associated random fields. The key step in our proof combines new moment and maximal inequalities, established by the authors for partial sums of multiindexed random variables, with the estimate of the convergence rate in the CLT for random fields under consideration.Comment: Published at http://dx.doi.org/10.1214/074921706000000167 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Central limit theorems for the excursion set volumes of weakly dependent random fields

The multivariate central limit theorems (CLT) for the volumes of excursion sets of stationary quasi-associated random fields on $\mathbb{R}^d$ are proved. Special attention is paid to Gaussian and shot noise fields. Formulae for the covariance matrix of the limiting distribution are provided. A statistical version of the CLT is considered as well. Some numerical results are also discussed.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ339 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm