This article describes the limiting distribution of the extremes of observations that arrive in clusters. We start by studying the tail behaviour of an individual cluster, and then we apply the developed theory to determine the limiting distribution of $\max\{X_j\,:\, j=0,\ldots, K(t)\}$ , where K(t) is the number of independent and identically distributed observations $(X_j)$ arriving up to the time t according to a general marked renewal cluster process. The results are illustrated in the context of some commonly used Poisson cluster models such as the marked Hawkes process.

The paper describes a new numerical method for the calculation of noncrossing probabilities for arbitrary boundaries by a Poisson process. We find the method to be simple in implementation, quick and efficient - it works reliably for Poisson processes of very high intensity n, up to several thousand. Hence, it can be used to detect unusual features in the finite-sample behaviour of empirical process and trace it down to very high sample sizes. It also can be used as a good approximation for noncrossing probabilities for Brownian motion and Brownian bridge, in particular when the boundaries are not regular. As a numerical example we demonstrate the divergence of normalized Kolmogorov-Smirnov statistics from their prescribed limiting distributions (Eicker (1979), Jaeshke (1979)) for quite large n in contrast to very regular behaviour of statistics of Mason (1983). For the Brownian motion case we considered square-root, Daniels' (1969) and Grooneboom's (1989) boundaries.

Johnson–Mehl tessellations can be considered as the results of spatial birth–growth processes. It is interesting to know when such a birth–growth process is completed within a bounded region. This paper deals with the limiting distributions of the time of completion for various models of Johnson–Mehl tessellations in ℝd and k-dimensional sectional tessellations, where 1 ≦ k < d, by considering asymptotic coverage probabilities of the corresponding Boolean models. Random fractals as the results of birth–growth processes are also discussed in order to show that a birth–growth process does not necessarily lead to a Johnson–Mehl tessellation.

Let Xn1 ≦ Xn2 ≦ ··· ≦ Xnn denote the order statistics from a sample of n independent, identically distributed random variables, and suppose that the variables Xnn, Xn, n–1, ···, when suitably normalized, have a non-trivial limiting joint distribution ξ1, ξ2, ···, as n → ∞. It is well known that the limiting distribution must be one of just three types. We provide a canonical representation of the stochastic process {ξn, n ≧ 1} in terms of exponential variables, and use this representation to obtain limit theorems for ξ n as n →∞.

An extremal-F process {Y (t); t ≧ 0} is defined as the continuous time analogue of sample sequences of maxima of i.i.d. r.v.'s distributed like F in the same way that processes with stationary independent increments (s.i.i.) are the continuous time analogue of sample sums of i.i.d. r.v.'s with an infinitely divisible distribution. Extremal-F processes are stochastically continuous Markov jump processes which traverse the interval of concentration of F. Most extremal processes of interest are broad sense equivalent to the largest positive jump of a suitable s.i.i. process and this together with known results from the theory of record values enables one to conclude that the number of jumps of Y (t) in (t1, t2] follows a Poisson distribution with parameter log t2/t1. The time transformation t→ et gives a new jump process whose jumps occur according to a homogeneous Poisson process of rate 1. This fact leads to information about the jump times and the inter-jump times. When F is an extreme value distribution the Y-process has special properties. The most important is that if F(x) = exp {—e–x} then Y(t) has an additive structure. This structure plus non parametric techniques permit a variety of conclusions about the limiting behaviour of Y(t) and its jump times.