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On records and related processes for sequences with trends

Part of: Stochastic processes Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

K. Borovkov
K. Borovkov*
Affiliation:
University of Melbourne
*
∗Postal address: Department of Mathematics and Statistics, University of Melbourne, Parkville 3052, Australia. Email address: [email protected].
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Abstract

We study the records and related variables for sequences with linear trends. We discuss the properties of the asymptotic rate function and relationships between the distribution of the long-term maxima in the sequence and that of a particular observation, including two characterization type results. We also consider certain Markov chains related to the process of records and prove limit theorems for them, including the ergodicity theorem in the regular case (convergence rates are given under additional assumptions), and derive the limiting distributions for the inter-record times and increments of records.

Keywords

Sequence with linear trendrecordsMarkov chains

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60J05: Discrete-time Markov processes on general state spaces 60K99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 3 , September 1999 , pp. 668 - 681
Copyright
Copyright © Applied Probability Trust 1999 

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Footnotes

Research supported by ARC Grant S69711423.

References

Balakrishnan, N. and Nevzorov, V. B. (1998). A record of records. In Handbook of Statistics. Vol. 16: Order Statistics: Theory and Methods, ed. Balakrishnan, N. and Rao, C. R. Elsevier, Amsterdam, pp. 515–570.Google Scholar
Ballerini, R. (1987). Another characterization of the type I extreme value distribution. Statist. Prob. Lett. 5, 83–85.Google Scholar
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, 801–828.CrossRefGoogle Scholar
Ballerini, R., and Resnick, S. I. (1987). Embedding sequences of successive maxima in extremal processes. J. Appl. Prob. 24, 827–837.Google Scholar
Bingham, N. H., Goldie, C. M., and Teugels, J. L. (1989). Regular variation, 2nd edn. CUP, Cambridge.Google Scholar
Borovkov, K., and Pfeifer, D. (1995). On record indices and record times. J. Statist. Plann. Inf. 45, 65–79.CrossRefGoogle Scholar
Bourbaki, N. (1976). Fonctions d'une variable réelle. (Théorieélémentaire). Hermann, Paris.Google Scholar
Diersen, J., and Trenkler, G. (1996). Records tests for trend in location. Statistics 28, 1–12.CrossRefGoogle Scholar
Embrechts, P., Kluppelberg, C., and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, New York.Google Scholar
Foster, F. G., and Stuart, A. (1954). Distribution-free tests in time-series based on the breaking the records. J. R. Statist. Soc. B 16, 1–22.Google Scholar
Foster, F. G., and Teichroew, D. (1955). A sampling experiment on the powers of the records tests for trend in a time series. J. R. Statist. Soc. B 17, 115–121.Google Scholar
Galambos, J. (1978). The Asymptotic Theory of Extreme Order Statistics. Wiley, New York.Google Scholar
de Haan, L., and Verkade, E. (1987). On extreme value theory in the presence of a trend. J. Appl. Prob. 24, 62–76.Google Scholar
Leadbetter, M. R., Lindgren, G., and Rootzen, H. (1983). Extremes and Related Properties of Random Sequences and Series. Springer, New York.Google Scholar
Lindvall, T. (1992). Lectures on the Coupling Method. Wiley, New York.Google Scholar
Nevzorov, V. B. (1987). Records. Theory Prob. Appl. 32, 201–228.Google Scholar
Pfeifer, D. (1989). Einführung in die Extremwertstatistik. Teubner, Stuttgart.Google Scholar
Pfeifer, D. (1997). A statistical model to analyse natural catastrophe claims by means of records values. Preprint No. 97-4. Institut für Mathematische Stochastik, University of Hamburg.Google Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.CrossRefGoogle Scholar
Robinson, M. E., and Tawn, J. A. (1995). Statistics for exceptional athletics records. Appl. Statist. 44, 499–511.CrossRefGoogle Scholar
Smith, R. L. (1988). Extreme value analysis of environmental time series: an application to trend detection in ground-level ozone. Theory in the presence of a trend. Statist. Sci. 4, 367–393.Google Scholar
Smith, R. L. (1988). Forecasting records by maximum likelihood. J. Am. Statist. Soc. 83, 331–338.Google Scholar
Smith, R. L., and Miller, J. E. (1986). A non-Gaussian state-space model and application to the prediction of records. J. R. Statist. Soc. B 48, 79–88.Google Scholar