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On the total time spent in records by a discrete uniform sequence

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Rudolf Grübel  and
Anke Reimers
Rudolf Grübel*
Affiliation:
Universität Hannover
Anke Reimers*
Affiliation:
Universität Hannover
*
∗ Postal address: Institut für Mathematische Stochastik, Universität Hannover, Postfach 60 09, D-30060 Hannover, Germany.
∗ Postal address: Institut für Mathematische Stochastik, Universität Hannover, Postfach 60 09, D-30060 Hannover, Germany.
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Abstract

We consider the sum Sd of record values in a sequence of independent random variables that are uniformly distributed on 1,…,d. This sum can be interpreted as the total amount of time spent in record lifetimes in the standard renewal theoretic setup. We investigate the distributional limit of Sd and some related quantities as d→∞. Some explicit values are given for d=6, a case that can be interpreted as a simple game of chance.

Keywords

Recordsuniform distributionperpetuities

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60K05: Renewal theory
Type
Short Communications
Information
Journal of Applied Probability , Volume 38 , Issue 3 , September 2001 , pp. 768 - 775
Copyright
Copyright © by the Applied Probability Trust 2001 

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References

Arnold, B. C., Balakrishnan, N., and Nagaraja, H. N. (1998). Records. John Wiley, New York.Google Scholar
Barbour, A. D., Holst, L., and Janson, S. (1992). Poisson Approximation. Oxford University Press.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.Google Scholar
Goldie, C. M. and Grübel, R. (1996). Perpetuities with thin tails. Adv. Appl. Prob. 28, 463–480.CrossRefGoogle Scholar
Grübel, R. (1994). The rank of the current lifetime. Statist. Prob. Lett. 20, 269–271.CrossRefGoogle Scholar
Grübel, R., and Reimers, A. (2001). On the number of iterations required by von Neumann addition. Theoret. Informatics Appl. 35, 187–206.CrossRefGoogle Scholar
Reimers, A. (2000). Rekorde in der Erneuerungstheorie. Doctoral Thesis, Universität Hannover.Google Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation and Point Processes. Springer, New York.Google Scholar
Rösler, U. (1991). A limit theorem for ‘Quicksort’. RAIRO Inform. Théor. Appl. 25, 85–100.Google Scholar
Scheffer, C. (1995). The rank of the present excursion. Stoch. Proc. Appl. 55, 101–108.Google Scholar
Scott, N. R. (1985). Computer Number Systems and Arithmetic. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar