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On the law of the iterated logarithm in the infinite variance case

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  09 April 2009

R. A. Maller
R. A. Maller
Affiliation:
Division of Mathematics and Statistics C.S.I.R.O. Melbourne and Perth, Australia
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Abstract

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The main purpose of the paper is to give necessary and sufficient conditions for the almost sure boundedness of (Sn – αn)/B(n), where Sn = X1 + X2 + … + XmXi being independent and identically distributed random variables, and αnand B(n) being centering and norming constants. The conditions take the form of the convergence or divergence of a series of a geometric subsequence of the sequence P(Sn − αn > a B(n)), where a is a constant. The theorem is distinguished from previous similar results by the comparative weakness of the subsidiary conditions and the simplicity of the calculations. As an application, a law of the iterated logarithm general enough to include a result of Feller is derived.

MSC classification

Secondary: 60F15: Strong theorems 60G50: Sums of independent random variables; random walks
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 30 , Issue 1 , September 1980 , pp. 5 - 14
Copyright
Copyright © Australian Mathematical Society 1980

References

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