A strong law of large numbers for vector Gaussian martingales and a statistical application in linear regression
type: Item
instance of: scholarly article
title: A strong law of large numbers for vector Gaussian martingales and a statistical application in linear regression
zbMATH Open document ID: 0613.60040
DOI: 10.1016/0167-7152(87)90029-0
author: Alain Le Breton
author: M. Musiela
published in: Statistics \& Probability Letters
publication date: 1987-01-01T00:00:00Z
review text: Let (\(\Omega\),\({\mathcal F},P)\) be a probability space and \(\{\) \({\mathcal F}_ t\), \(t\geq 0\}\) be an increasing family of \(\sigma\)-algebras such that \({\mathcal F}_ t={\mathcal F}_{t+0}\) and \({\mathcal F}_ 0\) contains all P-null sets. If \(\{X_ t,{\mathcal F}_ t\), \(t\geq 0\}\) is a right continuous d- dimensional Gaussian martingale such that \(X_ 0=0\), a.e., let \(<X>_ t=E(X_ tX^*_ t)\) where \(X^*_ t\) is the transpose of \(X_ t\). Since \(\{X_ tX^*_ t,{\mathcal F}_ t\), \(t\geq 0\}\) is a submartingale under matrix ordering, the \(<X>_ t\) is an increasing sequence of matrices. Let \(<X>^+_ t\) be the generalized inverse of \(<X>_ t\). As a consequence of the strong law of large numbers for real martingales, the authors show that (i) \(<X>^+_ t X_ t\to a\) limit with probability 1, as \(t\to \infty\) and (ii) the limit is zero iff \(\lim_{t\to \infty}trace <X>^+_ t=0\). As an application they discussed the strong consistency of an estimator of a (vector) parameter in a linear regression model. [The authors call \(<X>_ t\) the ''tensor increasing process'' of the \(X_ t\)-process. In martingale theory this term usually refers to the unique increasing process in the Doob-Meyer decomposition and need not be a constant except when the \(X_ t\)-process is Brownian motion. The assumption of nonrandomness was implied by the proof but not stated explicitly.]
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 62J05
Mathematics Subject Classification ID: 62M10
zbMATH DE Number: 3992549
zbMATH Keywords: Gaussian martingale
zbMATH Keywords: strong law of large numbers for real martingales
zbMATH Keywords: strong consistency of an estimator
zbMATH Keywords: linear regression model
zbMATH Keywords: Doob-Meyer decomposition
reviewed by: Malempati M. Rao
MaRDI profile type: Publication
full work available at URL: 0167-7152(87)90029-0
OpenAlex ID: W2015595077
cites work: Strong consistency of least squares estimates in normal linear regression
cites work: Calcul stochastique et problèmes de martingales
cites work: A strong law of large numbers for local martingales
description: scientific article; zbMATH DE number 3992549