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A pair of complementary theorems on convergence rates in the law of large numbers

Published online by Cambridge University Press:  24 October 2008

C. C. Heyde  and
V. K. Rohatgi
C. C. Heyde
Affiliation:
University of Sheffield
V. K. Rohatgi
Affiliation:
University of Sheffield and Michigan State University
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Extract

Introduction. Let Xi (i= 1, 2, 3,…) be a sequence of independent and identically distributed random variables with law ℒ(X) and write The Kolmogorov-Marcinkiewicz strong law of large numbers (Loève(6), p. 243) has the following statement:

If E|X|r < ∞, then with cr = 0 or EX according as r 1 or r ≥ 1.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 63 , Issue 1 , January 1967 , pp. 73 - 82
Copyright
Copyright © Cambridge Philosophical Society 1967

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References

REFERENCES

(1)Baum, L. E. & Katz, M.Convergence rates in the law of large numbers. Bull. Amer. Math. Soc. 69 (1963), 771–772.CrossRefGoogle Scholar
(2)Baum, L. E. & Katz, M.Convergence rates in the law of large numbers. Trans. Amer. Math. Soc. 120 (1965), 108–123.CrossRefGoogle Scholar
(3)Erdös, P.On a theorem of Hsu and Robbins. Ann. Math. Statist. 20 (1949), 286–291.CrossRefGoogle Scholar
(4)Karamata, J.Sur un mode de croissance régulière des fonctions. Mathematics (Cluj) 4 (1930), 38–53.Google Scholar
(5)Katz, M.The probability in the tail of a distribution. Ann. Math. Statist. 34 (1963), 312–318.CrossRefGoogle Scholar
(6)Loève, M.Probability theory. Van Nostrand, New York (1960).Google Scholar
(7)Spitzer, F.A combinatorial lemma and its application to probability theory. Trans. Amer. Math. Soc. 82 (1956), 323–339.CrossRefGoogle Scholar