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Simple approximations to the expected waiting time for a cluster of any given size, for point processes

Published online by Cambridge University Press:  01 July 2016

Ester Samuel-Cahn*
Affiliation:
Hebrew University
*
Postal address: Department of Statistics, The Hebrew University of Jerusalem, Jerusalem, Israel.

Abstract

For point processes, such that the interarrival times of points are independently and identically distributed, let T(L, m) denote the time until at least points cluster within an interval of length at most L. Let τ (L, m) + 1 be the total number of points observed until the above happens. Simple approximations of Eτ (L, m) and ET(L, m) are derived, as well as lower and upper bounds for their value. Approximations to the variances are also given. In particular the Poisson, Bernoulli and compound Poisson processes are discussed in detail. Some numerical tables are included.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 1983 

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Footnotes

This research was done while the author spent a sabbatical year at Columbia and Rutgers Universities.

References

Feller, W. (1968) An Introduction to Probability Theory and its Applications, Vol. 1, 3rd edn. Wiley, New York.Google Scholar
Glaz, J. (1979) Expected waiting time for the visual response. Biol. Cybernetics 35, 3941.Google Scholar
Glaz, J. (1981) Clustering of events in a stochastic process. J. Appl. Prob. 18, 268275.Google Scholar
Huntington, R. J. (1974) Distributions and Expectations for Clusters in Continuous and Discrete Cases, with Applications. , Rutgers University.Google Scholar
Huntington, R. J. (1976) Mean recurrence times for k successes within m trials. J. Appl. Prob. 13, 604607.Google Scholar
Huntington, R. T. and Naus, J. I. (1975) A simpler expression for kth nearest neighbor coincidence probabilities. Ann. Prob. 3, 894896.Google Scholar
Karlin, S. and Taylor, H. M. (1975) A First Course in Stochastic Processes, 2nd edn. Academic Press, New York.Google Scholar
Naus, J. (1974) Probabilities for a generalized birthday problem. J. Amer. Statist. Assoc. 69, 810815.Google Scholar
Naus, J. (1979) An indexed bibliography of clusters, clumps and coincidences. Internat. Statist. Rev. 47, 4778.Google Scholar
Naus, J. (1982) Approximations for distributions of scan statistics. J. Amer. Statist. Assoc. 77, 177183.CrossRefGoogle Scholar
Neff, N. D. and Naus, J. I. (1980) The Distribution of the Size of the Maximum Cluster of Points on a Line. IMS Series of Selected Tables in Mathematical Statistics VI. American Mathematical Society, Providence, RI.Google Scholar
Newell, G. F. (1963) Distribution for the smallest distance between any pair of kth nearest neighbor random points on a line. In Proc. Symp. Time Series Analysis, ed. Rosenblatt, M., Wiley, New York, 89103.Google Scholar
Saperstein, B. (1975) Note on a clustering problem. J. Appl. Prob. 12, 629632.Google Scholar