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On discrete time extremal processes

Published online by Cambridge University Press:  01 July 2016

R. W. Shorrock
R. W. Shorrock*
Affiliation:
Université de Montréal
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Abstract

Upper record values and times and inter-record times are studied in their rôles as embedded structures in discrete time extremal processes. Various continuous time approximations to the discrete-time processes are analysed, especially as processes over their state spaces. Discrete time processes, suitably normalized after crossing a threshold T, are shown to converge to limiting continuous time processes as T → ∞ under suitable assumptions on the underlying CDF F, for example, when 1 — F varies regularly at ∞, and more generally. Discrete time extremal processes viewed as processes over their state spaces are noted to have an interesting interpretation in terms of processes of population growth.

Keywords

RECORDSPOPULATION GROWTHINDEPENDENT INCREMENT POINT PROCESSLIMIT LAWS
Type
Research Article
Information
Advances in Applied Probability , Volume 6 , Issue 3 , September 1974 , pp. 580 - 592
Copyright
Copyright © Applied Probability Trust 1974 

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References

[1] Balkema, A. A. and de Hann, L. (1972) Residual lifetime at great age. Technical Report No. 36, Stanford University.Google Scholar
[2] Chung, K. L. (1968) A Course in Probability Theory. Harcourt, Brace and World, New York.Google Scholar
[3] Feller, W. (1966) An Introduction to Probability Theory and its Applications. Vol. II. John Wiley, New York.Google Scholar
[4] Kendall, D. G. (1948) On some models of population growth leading to Fisher's logarithmic series distribution. Biometrika 35, 6–15.Google Scholar
[5] Resnick, S. I. and Rubinovitch, M. (1973) The structure of extremal processes. Adv. Appl. Prob. 5, 287–307.Google Scholar
[6] Resnick, S. I. (1973) Extremal processes and record value times. J. Appl. Prob. 10, 864–868.Google Scholar
[7] Resnick, S. I. (1973) Inverses of extremal processes. Technical Report No. 47 Technical Report No. 47, Department of Statistics, Stanford University.Google Scholar
[8] Shorrock, R. (1972) A limit theorem for inter-record times. J. Appl. Prob. 9, 219–223.Google Scholar
[9] Shorrock, R. (1972) On record values and record times. J. Appl. Prob. 9, 316–326.Google Scholar
[10] Shorrock, R. (1973) Record values and inter-record times. J. Appl. Prob. 10, 543–555.Google Scholar
[11] Dwass, M. (1973) Extremal Processes– III. Discussion Paper No. 41. The Center for Mathematical studies in Economics and Management Science, Northwestern University.Google Scholar