Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-17T04:07:01.887Z Has data issue: false hasContentIssue false
  • English
  • Français

Dependence structure of Poisson-paced records

Part of: Distribution theory Stochastic processes

Published online by Cambridge University Press:  14 July 2016

J. A. Bunge  and
H. N. Nagaraja
J. A. Bunge*
Affiliation:
Cornell University
H. N. Nagaraja*
Affiliation:
The Ohio State University
*
∗Postal address: Statistics Center, Cornell University, Ithaca, NY 14853, USA.
∗∗Postal address: Department of Statistics, The Ohio State University, 1958 Neil Avenue, Columbus, OH 43210, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

Let Y0, Y1, Y2, ··· be an i.i.d. sequence of random variables with absolutely continuous distribution function F, and let {N(t), t ≧ 0} be a Poisson process with rate λ (t) and mean Λ(t), independent of the Yj's. We associate Y0 with the point t = 0, and Yj with the jth point of N(·), j ≧ 1. The first Yj (j ≧ 1) to exceed all previous ones is the first record value, and the time of its occurrence is the first record time; subsequent record values and times are defined analogously. For general Λ, we give the joint distribution of the values and times of the first n records to occur after a fixed time T, 0 ≦ T < ∞. Assuming that F satisfies Von Mises regularity conditions, and that λ (t)/Λ (t) → c ∈ (0, ∞) as t → ∞, we find the limiting joint p.d.f. of the values and times of the first n records after T, as T → ∞. In the course of this we correct a result of Gaver and Jacobs (1978). We also consider limiting marginal and conditional distributions. In addition, we extend a known result for the limit as the number of recordsK → ∞, and we compare the results for the limit as T → ∞ with those for the limit as K → ∞.

Keywords

RECORD VALUESRECORD TIMESINTERRECORD TIMESPOISSON PROCESSMARKOV CHAIN

MSC classification

Primary: 62E15: Exact distribution theory
Secondary: 60G55: Point processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 3 , September 1992 , pp. 587 - 596
Copyright
Copyright © Applied Probability Trust 1992 

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

Bunge, J. and Nagaraja, H. (1992) Exact distribution theory for some point process models. Adv. Appl. Prob. 24, 20–44.Google Scholar
Deheuvels, P. (1982) Spacings, record times and extremal processes. In Exchangeability in Probability and Statistics, ed. Koch, G. and Spizzichino, F., pp. 233–243. North-Holland, Amsterdam.Google Scholar
Deheuvels, P. (1983) The strong approximation of extremal processes (II). Z. Wahrscheinlichkeitsth. 62, 7–15.Google Scholar
Deheuvels, P. (1988) Strong approximations of k-th records and k-th record times by Wiener processes. Prob. Th. Rel. Fields 77, 195–209.Google Scholar
Dwass, M. (1964) Extremal processes. Ann. Math. Statist. 35, 1718–1725.Google Scholar
Galambos, J. (1987) The Asymptotic Theory of Extreme Order Statistics, 2nd edn. Kreiger, Malabar, Fl.Google Scholar
Gaver, D. (1976) Random record models. J. Appl. Prob. 13, 538–547.Google Scholar
Gaver, D. and Jacobs, P. (1978) Nonhomogeneously paced random records and associated extremal processes. J. Appl. Prob. 15, 543–551.Google Scholar
Nagaraja, H. N. (1988) Record values and related statistics - a review. Comm. Stat. - Theory Meth. 17, 2223–2238.Google Scholar
Nevzorov, V. (1987) Records. Theory Prob. Appl. 32, 201–228.Google Scholar
Pfeifer, D. (1987) On a joint strong approximation theorem for record and interrecord times. Probab. Th. Rel. Fields 75, 213–221.Google Scholar
Pickands, J. III (1971) The two dimensional Poisson process and extremal processes. J. Appl. Prob. 8, 745–756.CrossRefGoogle Scholar
Resnick, S. (1973) Limit laws for record values. Stoch. Proc. Appl. 1, 67–82.Google Scholar
Resnick, S. (1987) Extreme Values, Regular Variation, and Point Processes. Springer-Verlag, New York.Google Scholar
Shorrock, R. W. (1972) A limit theorem for inter-record times. J. Appl. Prob. 9, 219–223.Google Scholar
Westcott, M. (1977) The random record model. Proc. R. Soc. London A 356, 529–547.Google Scholar