On the strong law of large numbers of multivariate martingales with random norming
This article examines the almost sure behaviour of \(A_ n^{-1} S_ n\), where \(S_ n\) is a mean 0 multivariate martingale and \(A_ n\) is a random norming matrix. The results extend those of \textit{H. Kaufmann} [Stochastic Processes Appl. 26, 73-85 (1987; Zbl 0632.60040)] for the case where \(A_ n\) is non-random. Theorems 2 and 3 establish conditions on the \(A_ n\) in terms of the martingale differences, \(x_ n= S_ n- S_{n- 1}\), and the least eigenvalue of \(A_ n\), \(\lambda(A_ n)\). If \(\lambda(A_ n)\) increases almost surely to infinity, if the \(A_ n\) are monotonically increasing (i.e. \(A_{n+1}' A_{n+1}- A_ n' A_ n\) are positive definite) and if either the sum of \(E\| A_{n- 1} x_ n\|^ 2\) is finite (Theorem 2) or the sum of \(E(\| A_{n- 1} x_ n\|^ 2\mid {\mathcal F}(x_ 1, x_ 2,\dots, x_ n))\) is finite almost surely (Theorem 3), then the strong law of large numbers holds, \(A_ n^{-1} S_ n\to 0\) almost surely. Theorem 4 establishes that for square integrable martingales we may take \[ A_ n= \sum^ n_{i= 1} E(x_ i' x_ i\mid {\mathcal F}(x_ 1, x_ 2,\dots, x_{i- 1})) \] (subject to some additional conditions). This extends Corollary 3 of \textit{T. L. Lai} and \textit{C. Z. Wei} [Ann. Stat. 10, No. 1, 154-166 (1982; Zbl 0649.62060)].
scientific article; zbMATH DE number 724752