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The maximum and mean of a random length sequence

Part of: Stochastic processes Special processes

Published online by Cambridge University Press:  14 July 2016

Peter J. Haas
Peter J. Haas*
Affiliation:
IBM Research Division
*
∗Postal address: IBM Research Division, Almaden Research Center, 650 Harry Road, San Jose, CA 95120–6099, USA.
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Abstract

We obtain a limit theorem for the joint distribution of the maximum value and sample mean of a random length sequence of independent and identically distributed random variables. This extends a previous bivariate convergence result for fixed length sequences and incidentally yields a new proof of Berman's classical limit theorem for the maximum value of a random number of random variables. Our approach uses a property of record time sequences and leads to probabilistically intuitive proofs. We also consider the partition of a finite interval into a random number of subintervals by the points of a non-delayed renewal process. Using the bivariate convergence result for random length sequences, we establish a limit theorem for the joint distribution of the number and maximum length of the subintervals as the interval length becomes large. This leads to limiting results for the ratio of the maximum to the mean subinterval length. Such results are of interest in connection with a simple model of parallel processing.

Keywords

ANSCOMBE'S THEOREMRENEWAL THEORYEXTREME VALUE THEORYPARALLEL PROCESSING

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60K05: Renewal theory
Type
Short Communications
Information
Journal of Applied Probability , Volume 29 , Issue 2 , June 1992 , pp. 460 - 466
Copyright
Copyright © Applied Probability Trust 1992 

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References

[1] Anscombe, F. J. (1952) Large sample theory of sequential estimation. Proc. Camb. Phil. Soc. 48, 600–607.CrossRefGoogle Scholar
[2] Berman, S. M. (1962) Limiting distribution of the maximum term in sequences of dependent random variables. Ann. Math. Statist. 33, 894–908.Google Scholar
[3] Billingsley, P. (1968) Probability and Measure, 2nd edn. Wiley, New York.Google Scholar
[4] Galambos, J. (1987) The Asymptotic Theory of Extreme Order Statistics, 2nd edn. Robert E. Krieger, Malabar, FL.Google Scholar
[5] Haas, P. J. (1989) Workload imbalance and parallel processing efficiency. IBM Research Report RJ 6936, San Jose, CA.Google Scholar
[6] Leadbetter, M. R., Lindgren, G. and Rootzen, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York.CrossRefGoogle Scholar
[7] Resnick, S. I. (1987) Extreme Values, Regular Variation, and Point Processes. Springer-Verlag, New York.CrossRefGoogle Scholar
[8] Ross, S. M. (1983) Stochastic Processes. Wiley, New York.Google Scholar
[9] Rossberg, H. J. (1965) Das asymptotische Unabhängigkeit der kleinsten and grössten Werte einer Stichprobe vom Stichprobenmittel. Math. Nachr. 28, 305–318.CrossRefGoogle Scholar
[10] Tiago De Oliveira, J. (1961) The asymptotical independence of the sample mean and the extremes. Rev. Fac. Sci. Lisboa, Ser. A 8, 299–310.Google Scholar