• Bernoulli
  • Volume 5, Number 3 (1999), 483-493.

A note on limit theorems for multivariate martingales

Uwe Küchler and Michael Sørensen

Full-text: Open access


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 multidimensional 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 occurring in connection with statistical inference for stochastic delay equations.

Article information

Bernoulli, Volume 5, Number 3 (1999), 483-493.

First available in Project Euclid: 27 February 2007

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

central limit theorem likelihood inference multivariate Gaussian diffusions stochastic delay equations weak law of large numbers


Küchler, Uwe; Sørensen, Michael. A note on limit theorems for multivariate martingales. Bernoulli 5 (1999), no. 3, 483--493.

Export citation


  • [1] Aldous, D.J. and Eagleson, G.K. (1978) On mixing and stability of limit theorems. Ann. Probab., 6, 325-331.
  • [2] Barndorff-Nielsen, O.E. and Sørensen, M. (1991) Information quantities in non-classical settings. Computational Statist. Data Anal., 12, 143-158.
  • [3] Barndorff-Nielsen, O.E. and Sørensen, M. (1994) A review of some aspects of asymptotic likelihood theory for stochastic processes. Internat. Statist. Rev., 62, 133-165.
  • [4] Chaleyat-Maurel, M. and Elie, L. (1981) Diffusions gaussiennes. Astérisque, 84/85, 255-279.
  • [5] Davis, M.H.A. (1977) Linear Estimation and Stochastic Control. London: Chapman & Hall.
  • [6] Dietz, H.M. (1992) A non-Markovian relative of the Ornstein-Uhlenbeck process and some of its local statistical properties. Scand. J. Statist., 19, 363-379.
  • [7] Dzhaparidze, K. and Spreij, P. (1993) The strong law of large numbers for martingales with deterministic quadratic variation. Stochastics and Stochastics Rep., 42, 53-65.
  • [8] Feigin, P.D. (1985) Stable convergence of semimartingales. Stochastic Process. Appl., 19, 125-134.
  • [9] Gushchin, A.A. and Küchler, U. (1997) Asymptotic properties of maximum likelihood estimators for a class of linear stochastic equations with time delay. Preprint, Humboldt University of Berlin.
  • [10] Hall, P. and Heyde, C.C. (1980) Martingale Limit Theory and Its Application. New York: Academic Press.
  • [11] Helland, I.S. (1982) Central limit theorems for martingales with discrete or continuous time. Scand. J. Statist., 9, 79-94.
  • [12] Hutton, J.E. and Nelson, P.I. (1984) A mixing and stable central limit theorem for continuous time martingales. Technical Report No. 42, Kansas State University.
  • [13] Kaufmann, H. (1987) On the strong law of large numbers for multivariate martingales. Stochastic Process. Appl., 26, 73-85.
  • [14] Le Breton, A. (1977) Parameter estimation in a linear stochastic differential equation. In Transactions of the 7th Prague Conference on Information Theory, Statistical Decision Function and Random Processes, Vol. A, pp. 353-366. Prague: Academia.
  • [15] Le Breton, A. (1984) Propriétés asymptotiques et estimation des paramètres pour les diffusions gaussiennes homogènes hypoelliptiques dans le cas purement explosif. C. R. Acad. Sci. Paris. Ser. I, 299, 185-188.
  • [16] Le Breton, A. and Musiela, M. (1982) Estimation des paramètres pour les diffusions gaussiennes homogènes hypoelliptiques. C. R. Acad. Sci. Paris. Ser. I, 294, 341-344.
  • [17] Le Breton, A. and Musiela, M. (1985) Some parameter estimation problems for hypoelliptic homogeneous Gaussian diffusions. Banach Center Publ., 16, 337-356.
  • [18] Le Breton, A. and Musiela, M. (1986) Une loi des grands nombres pour les martingales locales continues vectorielles et son application en régression linéaire stochastique. C. R. Acad. Sci. Paris. Ser. I, 303, 421-424.
  • [19] Le Breton, A. and Musiela, M. (1987) A strong law of large numbers for vector Gaussian martingales and a statistical application in linear regression. Statist. Probab. Lett., 5, 71-73.
  • [20] Le Breton, A. and Musiela, M. (1989) Laws of large numbers for semimartingales with application to stochastic regression. Probab. Theory Related Fields, 81, 275-290.
  • [21] Lépingle, D. (1978) Sur le comportement asymptotique des martingales locales. In C. Dellacherie, P.A. Meyer, and M. Weil (eds), Séminaire de Probabilités XII, Lecture Notes in Mathematics 649. Berlin. Springer-Verlag.
  • [22] Liptser, R.Sh. (1980) A strong law of large numbers for local martingales. Stochastics, 3, 217-228.
  • [23] Melnikov, A.V. (1986) The law of large numbers for multidimensional martingales. Soviet Math. Dokl., 33, 131-135.
  • [24] Melnikov, A.V. and Novikov, A.A. (1990) Statistical inferences for semimartingale regression models. In B. Grigelionis, Yu.V. Prohorov, V.V. Sazonov and V. Statulevicius (eds), Probability Theory and Mathematical Statistics, Proceedings of the Fifth Vilnius Conference, Vol. 2, pp. 150-167. Utrecht: VSP.
  • [25] Rényi, A. (1958) On mixing sequences of sets. Acta Math. Acad. Sci. Hungar., 9, 215-228.
  • [26] Rényi, A. (1963) On stable sequences of events. Sankhya, Ser. A, 25, 293-302.
  • [27] Sørensen, M. (1991) Likelihood methods for diffusions with jumps. In N.U. Prabhu and I.V. Basawa (eds), Statistical Inference in Stochastic Processes, pp. 67-105. New York: Marcel Dekker.
  • [28] Stockmarr, A. (1996) Limits of autoregressive processes with a special emphasis on relations to cointegration theory. Ph.D. thesis, University of Copenhagen.