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]**H. Cramér (1946),*Mathematical methods of statistics*, Princeton Math. Ser., vol. 9, Princeton Univ. Press, Princeton, N.J. MR**8**, 39. MR**0016588 (8:39f)****[2]**C. G. Esseen (1945),*Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law*, Acta Math.**77**, 1-125. MR**7**, 312. MR**0014626 (7:312a)****[3]**M. L. Katz (1963),*Note on the Berry-Esseen theorem*, Ann. Math. Statist.**34**, 1107-1108. MR**27**# 1977. MR**0151996 (27:1977)****[4]**V. V. Petrov (1965),*An estimate of the deviation of the distribution of a sum of independent random variables from the normal law*, Dokl. Akad. Nauk SSSR**160**, 1013-1015 = Soviet Math. Dokl.**6**, 242-244. MR**31**#2754. MR**0178497 (31:2754)**

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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