A generalization of Kolmogorov's law of the iterated logarithm

Author:
R. J. Tomkins

Journal:
Proc. Amer. Math. Soc. **32** (1972), 268-274

MSC:
Primary 60F05

MathSciNet review:
0292142

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Abstract: A version of the law of the iterated logarithm is proved for sequences of independent random variables which satisfy the central limit theorem in such a way that the convergence of the appropriate moment-generating functions to that of the standard normal distribution occurs at a particular rate. Kolmogorov's law of the iterated logarithm is a corollary of this theorem which, unlike Kolmogorov's result, does not require boundedness of the random variables. Some iterated logarithm results for weighted averages of independent random variables are shown to follow from the main result. Moreover, some applications to sequences of independent, generalized Gaussian random variables are provided.

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DOI:
https://doi.org/10.1090/S0002-9939-1972-0292142-1

Keywords:
Independent random variables,
law of the iterated logarithm,
central limit theorem,
Kolmogorov's theorem,
weighted average,
Laplace distribution,
generalized Gaussian

Article copyright:
© Copyright 1972
American Mathematical Society