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A converse to the log-log law for Martingales

Published online by Cambridge University Press:  09 April 2009

W. L. Steiger
W. L. Steiger
Affiliation:
Centre de Recherches Mathématiques, Université de Montréal, Canada
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For sums of independent and identically distributed random variables xn, the HartmanWintner law of the iterated logarithm is equivalent to xn ∈ L2. We show that this is also true when the xn, form a stationary, ergodic martingale difference sequence. This is accomplished by extending a theorem of Volker Strassen to the present context.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 17 , Issue 4 , June 1974 , pp. 496 - 499
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Billingsly, P., ‘The Lindeberg-Lévy theorem for martingale’, Proc. Amer. Math. Soc. 12 (1961), 788–792.Google Scholar
[2]Hartman, P. and Wintner, A., ‘On the law of the iterated logarithm’, Amer. J. Math. 63 (1941), 169–176.CrossRefGoogle Scholar
[3]Steiger, W. L. and Zaremba, S. K., ‘The converse of the Hartman-Wintner theorem’, Z. Wahrscheinlickeitstheorie verw. Geb. 22 (1972), 193–194.CrossRefGoogle Scholar
[4]Stout, W. F., ‘A martingale analogue of Kolmogorov's law of the iterated logarithm’, Z. Wahrscheinlichkeilstheorie verw. Geb. 15 (1970), 279–290.Google Scholar
[5]Stout, W. F., ‘The Hartman-Wintner law of the iterated logarithm for martingales’, Ann. Math. Statist. 41 (1970), 2158–2160.Google Scholar
[6]Strassen, V., ‘A converse to the law of the of the iterated logarithm’, Z. Wahrscheinlichkeitstheorie verw. Geb. 4 (1966), 265–268.CrossRefGoogle Scholar