Some iterated logarithm results related to the central limit theorem.

Tomkins, R. J.

doi:10.1090/S0002-9947-1971-0275503-X

Some iterated logarithm results related to the central limit theorem.
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by R. J. Tomkins

Trans. Amer. Math. Soc. 156 (1971), 185-192

DOI: https://doi.org/10.1090/S0002-9947-1971-0275503-X

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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.

References

Andrew C. Berry, The accuracy of the Gaussian approximation to the sum of independent variates, Trans. Amer. Math. Soc. 49 (1941), 122–136. MR 3498, DOI 10.1090/S0002-9947-1941-0003498-3
Kai Lai Chung, A course in probability theory, Harcourt, Brace & World, Inc., New York, 1968. MR 0229268
W. Feller, Generalization of a probability limit theorem of Cramér, Trans. Amer. Math. Soc. 54 (1943), 361–372. MR 9262, DOI 10.1090/S0002-9947-1943-0009262-5
William Feller, An introduction to probability theory and its applications. Vol. I, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1957. 2nd ed. MR 0088081
William Feller, An introduction to probability theory and its applications. Vol. II, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0210154
Philip Hartman and Aurel Wintner, On the law of the iterated logarithm, Amer. J. Math. 63 (1941), 169–176. MR 3497, DOI 10.2307/2371287
Philip Hartman, Normal distributions and the law of the iterated logarithm, Amer. J. Math. 63 (1941), 584–588. MR 4405, DOI 10.2307/2371372
A. Kolmogoroff, Über das Gesetz des iterierten Logarithmus, Math. Ann. 101 (1929), no. 1, 126–135 (German). MR 1512520, DOI 10.1007/BF01454828
Michel Loève, Probability theory, 3rd ed., D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1963. MR 0203748

Remarque sur la loi du logarithme itéré

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V. V. Petrov, A bound for the deviation of the distribution of a sum of independent random variables from the normal law, Dokl. Akad. Nauk SSSR 160 (1965), 1013–1015 (Russian). MR 0178497
V. V. Petrov, On a relation between an estimate of the remainder in the central limit theorem and the law of iterated logarithm, Teor. Verojatnost. i Primenen. 11 (1966), 514–518 (Russian, with English summary). MR 0212855
Valentin V. Petrov, On the law of the iterated logarithm without assumptions about the existence of moments, Proc. Nat. Acad. Sci. U.S.A. 59 (1968), 1068–1072. MR 228052, DOI 10.1073/pnas.59.4.1068
V. Strassen, An invariance principle for the law of the iterated logarithm, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 3 (1964), 211–226 (1964). MR 175194, DOI 10.1007/BF00534910
Mary Weiss, On the law of the iterated logarithm, J. Math. Mech. 8 (1959), 121–132. MR 0102853, DOI 10.1512/iumj.1959.8.58008