A remark on the strong law of large numbers

Author:
Robert Chen

Journal:
Proc. Amer. Math. Soc. **61** (1976), 112-116

MSC:
Primary 60F15

DOI:
https://doi.org/10.1090/S0002-9939-1976-0420802-1

MathSciNet review:
0420802

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be mutually independent random variables such that and for all . For each let ; then, by the Kolmogorov criterion for mutually independent random variables, almost surely as for any positive constant . A deeper understanding of this theorem will be facilitated if we know the order of magnitude of as , where is the integer-valued random variable defined by . The present note does the work for a wide class of random variables by using Esseen's theorem and Katz-Petrov's theorem.

**[1]**Harald Cramér,*Mathematical Methods of Statistics*, Princeton Mathematical Series, vol. 9, Princeton University Press, Princeton, N. J., 1946. MR**0016588****[2]**Carl-Gustav Esseen,*Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law*, Acta Math.**77**(1945), 1–125. MR**0014626**, https://doi.org/10.1007/BF02392223**[3]**Melvin L. Katz,*Note on the Berry-Esseen theorem*, Ann. Math. Statist.**34**(1963), 1107–1108. MR**0151996**, https://doi.org/10.1214/aoms/1177704037**[4]**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**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1976-0420802-1

Keywords:
Esseen's theorem,
Euler-Maclaurin sum formula,
Katz-Petrov's theorem,
Kolmogorov's criterion,
mutually independent random variables,
strong law of large numbers

Article copyright:
© Copyright 1976
American Mathematical Society