DEPENDENCE STRUCTURE OF POISSON-PACED RECORDS

Let Y0, Y1, Y2, . . . be an i.i.d. sequence of random variables with absolutely continuous distribution function F, and let {N(t), t greater-than-or-equal-to 0} be a Poisson process with mte lambda(t) and mean LAMBDA(t), independent of the Y(j)'s. We associate Y0 with the point t = 0, and Y(j) with the jth point of N(.), j greater-than-or-equal-to 1. The first Y(j) (j greater-than-or-equal-to 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 LAMBDA, we give the joint distribution of the values and times of the first n records to occur after a fixed time T, 0 greater-than-or-equal-to T < infinity. Assuming that F satisfies Von Mises regularity conditions, and that lambda(t)/LAMBDA(t) --> c is-an-element-of (0, infinity) as t --> infinity, we find the limiting joint p.d.f. of the values and times of the first n records after T, as T --> infinity. 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 records K --> infinity, and we compare the results for the limit as T --> infinity with those for the limit as K --> infinity.