Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-05T15:25:03.220Z Has data issue: false hasContentIssue false
  • English
  • Français

Record values and inter-record times

Published online by Cambridge University Press:  14 July 2016

R. W. Shorrock
R. W. Shorrock*
Affiliation:
Université de Montréal
Get access Rights & Permissions [Opens in a new window]

Abstract

First, asymptotic results for inter-record times when the CDF of the underlying IID process is not necessarily continuous are obtained, by a stochastic order argument, from known results for the continuous case. Then the asymptotic behaviour of the bivariate process of upper-record values and inter-record times is studied. Finally, assuming continuity of the underlying CDF, we derive the law of the process of total times spent in sets of states, viewing upper record values as states and inter-record times as times spent in a state, the process so viewed being a discrete time continuous state Markov jump process.

The possible relevance of this result to single lane road traffic flow is indicated.

Keywords

RECORD VALUESINTER-RECORD TIMESLOG SERIES LAWCOMPOUND POISSONTRAFFIC FLOWCARAVAN
Type
Research Papers
Information
Journal of Applied Probability , Volume 10 , Issue 3 , September 1973 , pp. 543 - 555
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.)

Footnotes

This paper is based on part of the author's Ph.D. dissertation, prepared under the direction of Professor Herbert Solomon, with the support of Federal Highway Administration Contract FH 11-6885 with Stanford University. It was written while the author was in the Department of Mathematics at the University of British Columbia.

References

[1] Holmes, P. T. and Strawderman, W. (1969) A note on the waiting times between record observations. J. Appl. Prob. 6, 711–714.Google Scholar
[2] Neuts, M. F. (1967) Waiting times between record observations. J. Appl. Prob. 4, 206–208.Google Scholar
[3] Pickands, J. (1971) The two-dimensional Poisson process and extremal processes. J. Appl. Prob. 8, 745–756.Google Scholar
[4] Rényi, A. (1962) Théorie des éléments saillant d'une suite d'observations. Colloquium on Combinatorial Methods in Probability Theory. Mathematisk Institut, Aarhus Universitet, Denmark. 104–115.Google Scholar
[5] Shorrock, R. W. (1972) A limit theorem for inter-record times. J. Appl. Prob. 9, 219–223.Google Scholar
[6] Shorrock, R. W. (1972) On record values and record times. J. Appl. Prob. 9, 316–326.Google Scholar
[7] Strawderman, W. and Holmes, P. T. (1970) On the law of the iterated logarithm for inter-record times. J. Appl. Prob. 7, 432–439.Google Scholar
[8] Tata, M. N. (1969) On outstanding values in a sequence of random variables. Z. Wahrscheinlichkeitsth. 12, 9–20.Google Scholar
[9] Resnick, S. I. (1973) Limit laws for record values. Stochastic Processes and their Applications 1, 67–82.Google Scholar