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Extreme values of independent stochastic processes

Published online by Cambridge University Press:  14 July 2016

Bruce M. Brown  and
Sidney I. Resnick
Bruce M. Brown
Affiliation:
La Trobe University
Sidney I. Resnick
Affiliation:
Stanford University
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Abstract

The maxima of independent Weiner processes spatially normalized with time scales compressed is considered and it is shown that a weak limit process exists. This limit process is stationary, and its one-dimensional distributions are of standard extreme-value type. The method of proof involves showing convergence of related point processes to a limit Poisson point process. The method is extended to handle the maxima of independent Ornstein–Uhlenbeck processes.

Keywords

BROWNIAN MOTIONPOISSON RANDOM MEASURESEXTREMAL PROCESSESWEAK CONVERGENCEDOUBLE EXPONENTIAL DISTRIBUTIONORNSTEIN–UHLENBECK PROCESS
Type
Research Papers
Information
Journal of Applied Probability , Volume 14 , Issue 4 , December 1977 , pp. 732 - 739
Copyright
Copyright © Applied Probability Trust 1977 

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Footnotes

*

Support provided by NSF Grant OIP 75–14513 while on leave from Stanford University. The hospitality of CSIRO, Division of Mathematics and Statistics, Canberra and the Department of Statistics, SGS, Australian National University is gratefully acknowledged.

References

[1] Bartlett, M. S. (1946) The large sample theory of sequential tests. Proc. Camb. Phil. Soc. 42, 239–244.Google Scholar
[2] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
[3] Breiman, L. (1968) Probability. Addison-Wesley, Menlo Park, California.Google Scholar
[4] Brown, M. (1970) A property of Poisson processes and its application to macroscopic equilibrium of particle systems. Ann. Math. Statist. 41, 1935–1941.Google Scholar
[5] Cramér, H. (1951) Mathematical Methods in Statistics. Princeton University Press, Princeton, N.J.Google Scholar
[6] Gnedenko, B. V. (1943) | Sur | la distribution | limite du terme maximum d'une serie aleatoire. Ann. Math. 44, 423–453.Google Scholar
[7] Haan, L. De (1970) On Regular Variation and Its Application to the Weak Convergence of Sample Extremes. MC Tract 32, Mathematisch Centrum, Amsterdam.Google Scholar
[8] Haan, L. De (1974) Equivalence classes of regularly varying functions. Stoch. Proc. Appl. 2, 243–259.Google Scholar
[9] Haan, L. De and Resnick, S. I. (1976) Limit theory for multivariate sample extremes. Z. Wahrscheinlichkeitsth. To appear.Google Scholar
[10] Pickands, J. (1971) The two-dimensional Poisson process and extremal processes. J. Appl. Prob. 8, 745–756.Google Scholar
[11] Resnick, S. I. (1975) Weak convergence to extremal processes. Ann. Prob. 3, 951–960.Google Scholar
[12] Jagers, P. (1974) Aspects of random measure and point processes. Advances in Probability 3, ed. Ney, and Port, . Marcel Dekker, New York.Google Scholar