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A note on limit theorems for multivariate martingales

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  • Küchler, Uwe
  • Sørensen, Michael M.

Abstract

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.

Suggested Citation

  • Küchler, Uwe & Sørensen, Michael M., 1998. "A note on limit theorems for multivariate martingales," SFB 373 Discussion Papers 1998,45, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    • Handle: RePEc:zbw:sfb373:199845
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    References listed on IDEAS

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    1. Barndorff-Nielsen, O. E. & Sorensen, M., 1991. "Information quantities in non-classical settings," Computational Statistics & Data Analysis, Elsevier, vol. 12(2), pages 143-158, September.
    2. Kaufmann, Heinz, 1987. "On the strong law of large numbers for multivariate martingales," Stochastic Processes and their Applications, Elsevier, vol. 26, pages 73-85.
    3. Le Breton, A. & Musiela, M., 1987. "A strong law of large numbers for vector gaussian martingales and a statistical application in linear regression," Statistics & Probability Letters, Elsevier, vol. 5(1), pages 71-73, January.
    4. Feigin, Paul D., 1985. "Stable convergence of semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 19(1), pages 125-134, February.
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