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. 1 Introduction The law of large numbers and the central limit theorem for martingales have proved very useful tools for obtaining asymptotic results about estimators of parameters in stochastic process models. However, the multivariate versions of these results published so far have been stated either with conditions that are too strict for many statistical applications, or in a generality that makes application to particular statistical models very difficult. In this paper we give multivariate ..