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Dependence structure of Poisson-paced records

Part of: Stochastic processes Distribution theory

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.
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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 

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