Ukrainian Mathematical Journal

, Volume 58, Issue 7, pp 1139–1143 | Cite as

Bounded law of the iterated logarithm for multidimensional martingales normalized by matrices

  • V. O. Koval’
Brief Communications


We investigate a bounded law of the iterated logarithm for matrix-normalized weighted sums of martingale differences in R d . We consider the normalization of matrices inverse to the covariance matrices of these sums by square roots. This result is used for the proof of the bounded law of the iterated logarithm for martingales with arbitrary matrix normalization.


Covariance Matrix Random Vector Covariance Matrice Weighting Coefficient Iterate Logarithm 
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  1. 1.
    A. Koval, “A new law of the iterated logarithm in Rd with application to matrix-normalized sums of random vectors,” J. Theor. Probab., 15, No. 1, 249–257 (2002).CrossRefMathSciNetGoogle Scholar
  2. 2.
    T. L. Lai, “Some almost sure convergence properties of weighted sums of martingale difference sequences,” in: Almost Everywhere Convergence. II, Academic Press, Boston (1991), pp. 179–190.Google Scholar
  3. 3.
    P. Lankaster, Theory of Matrices, Academic Press, New York (1969).Google Scholar
  4. 4.
    R. Wittmann, “A general law of iterated logarithm,” Z. Wahrscheinlichkeitstheor. Verw. Geb., 68, No. 4, 521–543 (1985).CrossRefMathSciNetGoogle Scholar
  5. 5.
    M. Duflo, Random Iterative Models, Springer, Berlin (1997).zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • V. O. Koval’
    • 1
  1. 1.Zhytomyr Technological UniversityZhytomyr

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