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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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Extract

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).

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

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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