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On the strong law of large numbers for multivariate martingales

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  • Kaufmann, Heinz

Abstract

The strong law of large numbers is considered for a multivariate martingale normed by a sequence of square matrices. In particular multivariate martingale extensions of the strong laws of Kolmogorov and Marcinkiewicz-Zygmund are presented. Convergence to zero in L[alpha] is obtained under the same conditions. Norming by powers of the covariance matrix is considered in detail. Results are further used to derive conditions for strong consistency of the least squares estimator in linear regression with multivariate responses. These conditions do not necessarily assume square integrability of errors. They become particularly simple for polynomially bounded regressors. Two examples are treated, including polynomial regression.

Suggested Citation

  • Kaufmann, Heinz, 1987. "On the strong law of large numbers for multivariate martingales," Stochastic Processes and their Applications, Elsevier, vol. 26, pages 73-85.
    • Handle: RePEc:eee:spapps:v:26:y:1987:i::p:73-85
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    Cited by:

    1. Chan, N.H. & Cheung, Simon K.C. & Wong, Samuel P.S., 2020. "Inference for the degree distributions of preferential attachment networks with zero-degree nodes," Journal of Econometrics, Elsevier, vol. 216(1), pages 220-234.
    2. R. M. Balan & Ioana Schiopu-Kratina, 2004. "Asymptotic Results with Generalized Estimating Equations for Longitudinal data II," RePAd Working Paper Series lrsp-TRS398, Département des sciences administratives, UQO.
    3. Valery Koval, 2002. "A New Law of the Iterated Logarithm in Rd with Application to Matrix-Normalized Sums of Random Vectors," Journal of Theoretical Probability, Springer, vol. 15(1), pages 249-257, January.
    4. 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.
    5. Dzhaparidze, K. & Spreij, P., 1989. "On SLLN for martingales with deterministic variation," Serie Research Memoranda 0079, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics.

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