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Waitingtimes between record observations

Published online by Cambridge University Press:  14 July 2016

Marcel F. Neuts
Marcel F. Neuts*
Affiliation:
Purdue University
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Abstract

If Δr denotes the waitingtime between the (r − 1)st and the rth upper record in a sequence of independent, identically distributed random variables with a continuous distribution, then it is shown that Δr satisfies the weak law of large numbers and a central limit theorem.

This theorem supplements those of Foster and Stuart and Rényi, who investigated the index Vr of the rth upper record.

Qualitatively the theorems establish the intuitive fact that for higher records, the waitingtime between the last two records outweighs even the total waitingtime for previous records. This explains also why the asymptotic normality of logVr is very inadequate for approximation purposes—Barton and Mallows.

Type
Short Communications
Information
Journal of Applied Probability , Volume 4 , Issue 1 , November 1967 , pp. 206 - 208
Copyright
Copyright © Applied Probability Trust 

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References

[1] Barton, D. E. and Mallows, C. L. (1965) Some aspects of the random sequence. Ann. Math. Statist. 36, 236–260.CrossRefGoogle Scholar
[2] Chandler, K. N. (1952) The distribution and frequency of record values. J.R. Statist. Soc. B 14, 220–223.Google Scholar
[3] Foster, F. G. and Stuart, A. (1954) Distribution-free tests in time series based on the breaking of records. J.R. Statist. Soc. B 16, 1–22.Google Scholar
[4] Karlin, S. (1966) A First Course in Stochastic Processes. Academic Press, New York and London.Google Scholar
[5] Rényi, A. (1962) Théorie des éléments saillants d'une suite d'observations. Colloq. Combinatorial Meth. Prob. Theory, Aarhus University, 104–155.Google Scholar