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On a unified approach to the law of the iterated logarithm for martingales

Published online by Cambridge University Press:  17 April 2009

P.G. Hall  and
C.C. Heyde
P.G. Hall
Affiliation:
Mathematical Institute, University of Oxford, St Giles, Oxford, England;
C.C. Heyde
Affiliation:
Division of Mathematics and Statistics, CSIRO, Canberra, ACT.
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Abstract

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There are two distinct approaches in the literature to framing a version of the law of the iterated logarithm for martingales. One involves norming by constants, using the martingale variance and the other involves norming by random variables, using the sums of conditional variances of the increments, given their past. In this paper a portmanteau approach is provided, still based on the Skorokhod representation of the martingale, but involving normalization by more general random variables. This extends the functional forms of all the previously existing results.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 14 , Issue 3 , June 1976 , pp. 435 - 447
Copyright
Copyright © Australian Mathematical Society 1976

References

[1]Heyde, C.C. and Scott, D.J., “Invariance principles for the law of the iterated logarithm for martingales and processes with stationary increments”, Ann. Probability 1 (1973), 428–436.CrossRefGoogle Scholar
[2]Jain, Naresh C., Jogdeo, Kumar and Stout, William F., “Upper and lower functions for martingales and mixing processes”, Ann. Probability 3 (1975), 119–145.CrossRefGoogle Scholar
[3]Lévy, Paul, Processus stochastiques et mouvement brownien (Suivi d'une note de M. Loève. Deuxième édition revue et augmentée. Gauthier-Villars, Paris, 1965).Google Scholar
[4]Neveu, Jacques, Mathematical foundations of the calculus of probability (translated by Feinstein, Amiel. Holden-Day, San Francisco, California; London; Amsterdam; 1965).Google Scholar
[5]Stout, William F., “A martingale analogue of Kolmogorov's law of the iterated logarithm”, Z. Wahrscheinlichkeitstheorie und verw. Gebiete 15 (1970), 279–290.CrossRefGoogle Scholar
[6]Strassen, V., “An invariance principle for the law of the iterated logarithm”, Z. Wahrscheinlichkeitstheorie und verw. Gebiete 3 (1964), 211–226.CrossRefGoogle Scholar
[7]Strassen, Volker, “Almost sure behavior of sums of independent random variables and martingales”, Proc. Fifth Berkeley Sympos. Math. Statistics Probability, Volume II, Part 1, 315–343 (University of California Press, Berkeley and Los Angeles, 1967).Google Scholar