A Note on Limit Theorems for Multivariate Martingales

Multivariate versions of the law of large numbers and the central limit theorem for martingales are given in a generality that is often necessary when studying statistical inference for stochastic process models. To illustrate the usefulness of the results, we consider estimation for a multi-dimensional Gaussian diffusion, where results on consistency and asymptotic normality of the maximum likelihood estimator are obtained in cases that were not covered by previously published limit theorems. The results are also applied to martingales of a different nature, which are typical of the problems occuring in connection with statistical inference for stochastic delay equations. Key words: Central limit theorem; multivariate Gaussian diffusions; likelihood inference; stochastic delay equations; weak law of large numbers. Research for this paper was partially carried out within Sonderforschungsbereich 373. This paper was printed using funds made available by the Deutsche Forschungsgemeinsc..