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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Some iterated logarithm results related to the central limit theorem.

Author: R. J. Tomkins
Journal: Trans. Amer. Math. Soc. 156 (1971), 185-192
MSC: Primary 60.30
MathSciNet review: 0275503
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Abstract: An iterated logarithm theorem is presented for sequences of independent, not necessarily bounded, random variables, the distribution of whose partial sums is related to the standard normal distribution in a particular manner. It is shown that if a sequence of independent random variables satisfies the Central Limit Theorem with a sufficiently rapid rate of convergence, then the law of the iterated logarithm holds. In particular, it is demonstrated that these results imply several known iterated logarithm results, including Kolmogorov’s celebrated theorem.

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Keywords: Independent random variables, law of the iterated logarithm, Central Limit Theorem, normal distribution, Berry-Esseen bounds, Lindeberg’s criterion, Kolmogorov’s theorem
Article copyright: © Copyright 1971 American Mathematical Society