Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-28T09:38:27.464Z Has data issue: false hasContentIssue false
  • English
  • Français

Extremal processes and record value times

Published online by Cambridge University Press:  14 July 2016

Sidney I. Resnick
Sidney I. Resnick*
Affiliation:
Stanford University
Get access Rights & Permissions [Opens in a new window]

Abstract

Let {Xn, n ≧ 1} be i.i.d. and Yn = max {X1,…, Xn}. Xj is a record value of {Xn} if Yj > Yj–1 The record value times are Ln, n ≧ 1 and inter-record times are Δn, n ≧ 1. The known limiting behavior of {Ln} and {Δn} is close to that of a non-homogeneous Poisson process and an explanation of this is obtained by embedding {Yn} in a suitable extremal process which jumps according to a non-homogeneous Poisson process.

Keywords

RECORD VALUERECORD TIMESPOISSON PROCESSEXTREMAL PROCESSMAXIMA
Type
Short Communications
Information
Journal of Applied Probability , Volume 10 , Issue 4 , December 1973 , pp. 864 - 868
Copyright
Copyright © Applied Probability Trust 1973 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Freedman, D. (1971) Markov Chains. Holden-Day, San Francisco.Google Scholar
[2] Holmes, P. T. and Strawderman, W. (1969) A note on the waiting times between record observations. J. Appl. Prob. 6, 711–714.CrossRefGoogle Scholar
[3] Neuts, M. F. (1967) Waiting times between record observations. J. Appl. Prob. 4, 206–208.CrossRefGoogle Scholar
[4] Pickands, J. (1971) The two-dimensional Poisson process and extremal processes. J. Appl. Prob. 8, 745–756.CrossRefGoogle Scholar
[5] Rényi, A. (1962) Théorie des elements saillant d'une suite d'observations. Colloquium on Combinatorial Methods in Probability Theory. Mathematisk Institut, Aarhus Universitet, Denmark. 104–115.Google Scholar
[6] Resnick, S. I. and Rubinovitch, M. (1973) The structure of extremal processes. Adv. Appl. Prob. 5, 287–307.CrossRefGoogle Scholar
[7] Shorrock, R. W. (1972) A limit theorem for inter-record times, J. Appl. Prob. 9, 219–223.CrossRefGoogle Scholar
[8] Shorrock, R. W. (1972) On record values and record times. J. Appl. Prob. 9, 316–326.CrossRefGoogle Scholar
[9] Strawderman, W. and Holmes, P. T. (1970) On the law of the iterated logarithm for inter-record times. J. Appl. Prob. 7, 432–439.CrossRefGoogle Scholar
[10] Tata, M. N. (1969) On outstanding values in a sequence of random variables. Z. Wahrscheinlichkeitsth. 12, 9–20.CrossRefGoogle Scholar