We study the weak convergence of the extremes of supercritical branching Lévy processes $\{\mathbb{X}_t, t \ge0\}$ whose spatial motions are Lévy processes with regularly varying tails. The result is drastically different from the case of branching Brownian motions. We prove that, when properly renormalized, $\mathbb{X}_t$ converges weakly. As a consequence, we obtain a limit theorem for the order statistics of $\mathbb{X}_t$.

An rth-order extremal process Δ(r) = (Δ(r)t)t≥0 is a continuous-time analogue of the rth partial maximum sequence of a sequence of independent and identically distributed random variables. Studying maxima in continuous time gives rise to the notion of limiting properties of Δt(r) as t ↓ 0. Here we describe aspects of the small-time behaviour of Δ(r) by characterising its upper and lower classes relative to a nonstochastic nondecreasing function bt > 0 with limt↓bt = 0. We are then able to give an integral criterion for the almost sure relative stability of Δt(r) as t ↓ 0, r = 1, 2, . . ., or, equivalently, as it turns out, for the almost sure relative stability of Δt(1) as t ↓ 0.

Pareto distributions are most popular for modeling heavy tailed data. Here, we obtain weak limits of a sequence of extremal and a sequence of additive processes constructed by a series of Bernoulli point processes with bivariate Pareto space components. For the limiting processes we derive the one dimensional distributions in explicit forms. Some of the main properties of these distributions are also proved.

This paper is devoted to the investigation of limit theorems for extremes with random sample size under general dependence-independence conditions for samples and random sample size indexes. Limit theorems of weak convergence type are obtained as well as functional limit theorems for extremal processes with random sample size indexes.

A study is made of the extremal process generated by i.i.d. random variables appearing at the events of a non-homogeneous Poisson process, 𝒫. If 𝒫 has an exponentially increasing rate function, then records eventually occur in a homogeneous Poisson process. The size of the latest record has a classical extreme value distribution.

Necessary and sufficient conditions are given for a distribution function in ℝ2 to be max-infinitely divisible. The d.f. F is max i.d. if Ft is a d.f. for every t > 0. This property is essential in defining multivariate extremal processes and arises in an approach to the study of the range of an i.i.d. sample.

Let {X(t),t ≧0} be a process with stationary independent increments which is stochastically continuous with right-continuous paths and normalized so that X(0)=0. Let Z1(t) = X(t), Z2(t) = sup0≦s≦tX(s) and Z3 (t) = largest positive jump of X in (0, t] if there is one; = 0 otherwise. Then for i = 1,2,3 and x > 0: limt↓0t—1P[Zi(t) > x] = M+(x) at all points of continuity of M+, the Lévy measure of X.

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.