A note on limit theorems for multivariate martingales. (English) Zbl 0943.60016

Küchler, Uwe; Sørensen, Michael

Bernoulli 5, No. 3, 483-493 (1999).

The main result is a central limit theorem for multivariate martingales under relatively mild assumptions. As a corollary, a weak law of large numbers is established under the same conditions. The hypotheses are sufficiently general as to allow these results to be applied in situations where previous theorems of this type cannot. As examples, the authors use the CLT to discern the asymptotic distribution of the maximum-likelihood estimator in Gaussian diffusion models and to make statistical inferences about the solutions of stochastic delay equations. Similarly, the weak law is used to study the consistency of least-squares estimators in certain semi-martingale models.

Reviewer: R.J.Tomkins (Regina)

MSC:

60F05

Central limit and other weak theorems

Keywords:

central limit theorem; weak law of large numbers; multivariate martingales; maximum likelihood; stochastic delay equations; semi-martingales; Gaussian diffusions