A generalization of Kolmogorov’s law of the iterated logarithm

Tomkins, R. J.

doi:10.1090/S0002-9939-1972-0292142-1

A generalization of Kolmogorov’s law of the iterated logarithm
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Proc. Amer. Math. Soc. 32 (1972), 268-274 Request permission

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.

References

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Remarque sur la loi du logarithme itéré

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Some results on almost sure and complete convergence in the independent and martingale cases

R. J. Tomkins, Some iterated logarithm results related to the central limit theorem, Trans. Amer. Math. Soc. 156 (1971), 185–192. MR 275503, DOI 10.1090/S0002-9947-1971-0275503-X