On the strong law of large numbers for multivariate martingales

AbstractThe strong law of large numbers is considered for a multivariate martingale normed by a sequence of square matrices. In particular multivariate martingale extensions of the strong laws of Kolmogorov and Marcinkiewicz-Zygmund are presented. Convergence to zero in Lα is obtained under the same conditions. Norming by powers of the covariance matrix is considered in detail. Results are further used to derive conditions for strong consistency of the least squares estimator in linear regression with multivariate responses. These conditions do not necessarily assume square integrability of errors. They become particularly simple for polynomially bounded regressors. Two examples are treated, including polynomial regression