Hostname: page-component-77c89778f8-n9wrp Total loading time: 0 Render date: 2024-07-17T04:12:28.412Z Has data issue: false hasContentIssue false
  • English
  • Français

On record and inter-record times for a sequence of random variables defined on a Markov chain

Published online by Cambridge University Press:  01 July 2016

Gary Lee Guthrie  and
Paul T. Holmes
Gary Lee Guthrie
Affiliation:
Clemson University
Paul T. Holmes
Affiliation:
Clemson University
Get access Rights & Permissions [Opens in a new window]

Abstract

The familiar three theorems of Rényi concerning the record times in an i.i.d. sequence of random variables are extended to the record times and inter-record times of a sequence of dependent, non-identically distributed random variables defined on a finite Markov chain. These theorems are the Central Limit Theorem (C.L.T.), the Strong Law of Large Numbers (S.L.L.N.) and the Law of the Iterated Logarithm (L.I.L.). Similar results are also obtained for m-record times, inter-m-record times, and for the continuous parameter situation when observations are taken at the epochs of a Poisson process.

Keywords

RECORD TIMESINTER RECORD TIMESMARKOV CHAINCENTRAL LIMIT THEOREMLAW OF LARGE NUMBERSLAW OF THE ITERATED LOGARITHMM-RECORDS
Type
Research Article
Information
Advances in Applied Probability , Volume 7 , Issue 1 , March 1975 , pp. 195 - 214
Copyright
Copyright © Applied Probability Trust 1975 

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] Cheong, C. K., de Smit, J. H. A. and Teugels, J. L. (1973) Bibliography on semi-Markov processes. C. O. R. E. Discussion Paper No. 7310. C.O.R. E., Heverlee, Belgium.Google Scholar
[2] Chung, K. L. (1968) A Course in Probability Theory. Harcourt, Brace and World, Inc., New York.Google Scholar
[3] Feller, W. (1968) An Introduction to Probability Theory and Its Applications. Vol. I, 3rd ed. John Wiley, New York.Google Scholar
[4] Holmes, P. T. and Strawderman, W. E. (1969) A note on the waiting times between record observations. J. Appl. Prob. 6, 711–714.CrossRefGoogle Scholar
[5] Neuts, M. F. (1967) Waiting times between record observations. J. Appl. Prob. 4, 206–208.Google Scholar
[6] Pickands, J. III. (1971) The two-dimensional Poisson process and extremal processes. J. Appl. Prob. 8, 745–756.Google Scholar
[7] Pyke, R. (1961) Markov renewal processes: definitions and preliminary properties. Ann. Math. Statist. 32, 1231–1242.CrossRefGoogle Scholar
[8] Pyke, R. and Shaufele, R. (1964) Limit theorems for Markov renewal processes. Ann. Math. Statist. 35, 1746–1764.Google Scholar
[9] Rényi, A. (1962) Théorie des elements saillants d'une suite d'observations. Colloquium on Combinatorial Methods in Probability Theory. Aarhus University. 104–115.Google Scholar
[10] Resnick, S. I. (1972) Stability of maxima of random variables defined on a Markov chain. Adv. Appl. Prob. 4, 285–295.Google Scholar
[11] Spitzer, F. (1964) Principles of Random Walk. D. Van Nostrand, Princeton.Google Scholar
[12] Strawderman, W. E. and Holmes, P. T. (1970) On the law of the iterated logarithm for inter-record times. J. Appl. Prob. 7, 432–439.Google Scholar