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On the asymptotic behaviour of sample spacings

Published online by Cambridge University Press:  24 October 2008

John Hawkes
John Hawkes
Affiliation:
University College of Swansea
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Summary

Suppose that we are given a random sample of size n chosen according to the uniform distribution on the unit interval. Let Zn(x) = Zn(x, ω) be the length of the unique left-closed and right-open sample spacing that contains x. The purpose of this paper is to examine the almost sure, and exceptional, growth rates of the process {Zn}. The typical maximum growth rate and the growth rate of the maximum can be of quite different orders of magnitude as is shown by the following two results.

Theorem 2. With probability one we have

for almost all x.

Theorem 3. With probability one we have

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 90 , Issue 2 , September 1981 , pp. 293 - 303
Copyright
Copyright © Cambridge Philosophical Society 1981

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References

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