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An extension of the law of the iterated logarithm

Published online by Cambridge University Press:  24 October 2008

J. W. S. Cassels
J. W. S. Cassels
Affiliation:
The UniversityManchester
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1. Let x1, x2, …, xn, … be a set of independent variables each with a uniform probability distribution in 0 ≤ x ≤ 1. If 0 ≤ α < β ≤ 1 we denote by FN (α, β) the number of x1, …, xN which satisfy α < x ≤ β,

and put RN(α, β) = FN(α, β) − N(β − α).

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 47 , Issue 1 , January 1951 , pp. 55 - 64
Copyright
Copyright © Cambridge Philosophical Society 1951

References

REFERENCES

(1)Cassels, J. W. S.Some metrical theorems of Diophantine approximation III, IV. Proc. Cambridge Phil. Soc. 46 (1950), 219–25 and Proc. K. Ned. Akad. Amsterdam, 53 (1950), 176–87 (= Indagationes Math. 12 (1950), 14–25).CrossRefGoogle Scholar
(2)Khintchine, A. Ya.Über einen Satz der Wahrscheinlichkeitsrechnung. Fund. Math. 6 (1924), 9–20.CrossRefGoogle Scholar
(3)Khintchine, A. Ya.Asymptotische Gesetze der Wahrscheinlichkeitsrechnung. Ergebn. Math, iv, 2 (1933), 65, Hilfssatz 3.Google Scholar