Strong limit theorems for quasi-orthogonal random fields
type: Item
instance of: scholarly article
title: Strong limit theorems for quasi-orthogonal random fields
zbMATH Open document ID: 0685.60033
DOI: 10.1016/0047-259X(89)90039-0
published in: Journal of Multivariate Analysis
publication date: 1989-01-01T00:00:00Z
review text: A double sequence \(\{X_{ik};i,k\geq 1\}\) of random variables is called quasi-orthogonal if a real double sequence \(\rho\) (m,n) exists such that \[ | E X_{ik}X_{j\ell}| \leq \rho (| i-j|,| k-\ell |)\sigma_{ik}\sigma_{j\ell}\quad for\quad all\quad i,j,k,\ell \geq 1, \] where \(\sigma^ 2_{ik}\) is the variance of \(X_{ik}\). For non- decreasing sequences \(\{\lambda_ 1(n)\}\) and \(\{\lambda_ 2(n)\}\), let \(\xi_{mn}=\sum^{m}_{i=1}\sum^{n}_{k=1}X_{ik}/(\lambda_ 1(m)\lambda_ 2(n)).\) Conditions are given which ensure that \(\xi_{mn}\to 0\) almost surely (a.s.) as \(m+n\to \infty.\) Similar results are proved for subsequences of \(\{\xi_{mn}\}\). In addition, sufficient conditions for \[ \lim_{n\to \infty}a_ n^{- 1}\sum_{(i,k)\in Q_ n}X_{ij}=0\quad a.s. \] are presented, where \(Q_ 1\subset Q_ 2\subset..\). are non-empty sets of lattice points in the first quadrant. The rates of these convergences are also explored.
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60G46
Mathematics Subject Classification ID: 60G60
zbMATH DE Number: 4122960
zbMATH Keywords: random field
zbMATH Keywords: quasi-orthogonality
zbMATH Keywords: rectangular mean
zbMATH Keywords: strong law of
zbMATH Keywords: large numbers
author: Ferenc Móricz
reviewed by: R. James Tomkins
MaRDI profile type: Publication
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full work available at URL: 0047-259x(89)90039-0
OpenAlex ID: W1966485635
description: scientific article; zbMATH DE number 4122960