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Embedding sequences of successive maxima in extremal processes, with applications

Published online by Cambridge University Press:  14 July 2016

Rocco Ballerini  and
Sidney I. Resnick
Rocco Ballerini*
Affiliation:
University of Florida
Sidney I. Resnick
Affiliation:
Colorado State University
*
∗Postal address: Department of Statistics, University of Florida, Gainesville, FL 32611, USA.
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Abstract

Consequences of embedding sequences {Mn} in an extremal-F process are discussed where Mn represents the maximum of n independent (but not necessarily identically distributed) random variables. Various limit theorems are proved for the sample record rate, record times, inter-record times, and record values. These results are illustrated with applications to three particular record models: the Yang (1975) record model where population size increases geometrically, a record model where linear improvement is present, and a record model incorporating features of the previous two.

Keywords

RECORDSGUMBEL DISTRIBUTION
Type
Research Papers
Information
Journal of Applied Probability , Volume 24 , Issue 4 , December 1987 , pp. 827 - 837
Copyright
Copyright © Applied Probability Trust 1987 

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Footnotes

This work was partially supported by a Research Development Award at the University of Florida.

∗∗

Present address: Cornell University, OR/IE, Upson Hall, Ithaca, NY14853, USA.

Initially supported by NSF Grant DMS 8202335 and at the end by a UK Science and Engineering Research Council Fellowship at Sussex University.

References

Ballerini, R. and Resnick, S. I. (1985) Records from improving populations. J. Appl. Prob. 22, 487–502.Google Scholar
Ballerini, R. and Resnick, S. I. (1987) Records in the presence of a linear trend. Adv. Appl. Prob. 19, 000–000.Google Scholar
Feller, W. (1968) An Introduction to Probability Theory and its Applications , Vol. 1, 3rd edn., Wiley, New York.Google Scholar
Haan, L. De (1970) On Regular Variation and its Application to the Weak Convergence of Sample Extremes. , Mathematical Centre Tracts 32, Mathematisch Centrum, Amsterdam.Google Scholar
Haan, L. De (1984) Extremal processes. In Statistical Extremes and Applications , ed. Tiago de Oliveira, J., Reidel, Dordrecht.Google Scholar
Loève, M. (1977) Probability Theory I , 4th Ed., Springer-Verlag, New York.Google Scholar
Resnick, S. I. (1974) Inverses of extremal processes. Adv. Appl. Prob. 6, 392–406.Google Scholar
Resnick, S. I. (1983) Extremal Processes. Encyclopedia of Statistical Sciences , ed. Johnson, N. L. and Kotz, S. Wiley, New York.Google Scholar
Resnick, S. I. and Rubinovitch, M. (1973) The structure of extremal processes. Adv. Appl. Prob. 5, 287–307.CrossRefGoogle Scholar
Smith, R. L. and Miller, J. (1984) Predicting records. Preprint, Imperial College, London.Google Scholar
Vervaat, W. (1972) Success Epochs in Bernoulli Trials with Applications in Number Theory. Mathematical Centre Tracts 42, Amsterdam.Google Scholar
Yang, M. C. K. (1975) On the distribution of the inter-record times in an increasing population. J. Appl. Prob. 12, 148–154.CrossRefGoogle Scholar