A strong law of large numbers for vector Gaussian martingales and a statistical application in linear regression
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.]
strong law of large numbers for real martingales
scientific article; zbMATH DE number 3992549