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 → ∞.

Let Y0, Y1, Y2, … be an i.i.d. sequence of random variables with continuous distribution function, and let P be a simple point process on 0≦t≦∞, independent of the Yj's. We assume that P has a point at t = 0; we associate Yj with the jth point of j≧0, and we say that the Yj's occur at the arrival times of P. Y0 is considered a ‘reference value'. The first Yj (j≧1) to exceed all previous ones is called the first ‘record value', and the time of its occurrence is the first ‘record time'. Subsequent record values and times are defined analogously. We give an infinite series representation for the joint characteristic function of the first n record times, for general P; in some cases the series can be summed. We find the intensity of the record process when P is a general birth process, and when P is a linear birth process with m immigration sources we find the distribution of the number of records in (0, t]. For m = 0 (the Yule process) we give moments of record times and a compact form for the record process intensity. We show that the records occur according to a homogeneous Poisson process when m = 1, and we display a different model with the same behavior, leading to statistical non-identifiability if only the record times are observed. For m = 2, the records occur according to a semi-Markov process; again we display a different model with the same behavior. Finally we give a new derivation of the joint distribution of the interrecord times when P is an arbitrary Poisson process. We relate this result to existing work and to the classical record model. We also obtain a new characterization of the exponential distribution.