In 1973, Williams [D. Williams, On Rényi's ‘record’ problem and Engel's series, Bull. London Math. Soc.5 (1973), 235–237] introduced two interesting discrete Markov processes, namely C-processes and A-processes, which are related to record times in statistics and Engel's series in number theory respectively. Moreover, he showed that these two processes share the same classical limit theorems, such as the law of large numbers, central limit theorem and law of the iterated logarithm. In this paper, we consider the large deviations for these two Markov processes, which indicate that there is a difference between C-processes and A-processes in the context of large deviations.

We study a random record model where the observation Xi has continuous distribution function Fαi (αi > 0) and the number of available observations is random and independent of the observations. We obtain the joint distribution of the record values and inter-record times for our model. We investigate the distribution of the number of records when the number of observations has one of the common distributions and the α's increase geometrically or linearly. A particularly interesting case arises when the observations arrive at time points paced by a Poisson point process. For this model we obtain distributional results for the inter-arrival times of records for a large class of combinations of α structures and intensity functions.

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.

Extreme-value theory is considered in the context of independent but not identically distributed random variables. The departure from the case of identical distributions comes from a trend added to i.i.d. observations. Unlike previous authors we consider trends that do not completely destroy the asymptotic behaviour from the i.i.d. case.

This note indicates two alternative proofs for a result of Galambos and Seneta [1] on the distribution of ratios of successive record times in a sequence of independent and identically distributed random variables. Some consequences of one method of proof are discussed, and its analogues in a continuous time setting are described.

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.

Let {Xn, n ≧ 1} be i.i.d. and Yn = max {X1,…, Xn}. Xj is a record value of {Xn} if Yj > Yj–1 The record value times are Ln, n ≧ 1 and inter-record times are Δn, n ≧ 1. The known limiting behavior of {Ln} and {Δn} is close to that of a non-homogeneous Poisson process and an explanation of this is obtained by embedding {Yn} in a suitable extremal process which jumps according to a non-homogeneous Poisson process.