Let, for each t ∈ T, ψ(t, ۔) be a random measure on theBorel σ-algebra in ℝd such that Eψ(t, ℝd)k < ∞ for all k and let $\widehat{\psi}$(t, ۔) beits characteristic function. We call the function$\widehat{\psi}$(t1,…, tl ; z1,…, zl) = ${\sf E}\prod^l_{j=1}\widehat{\psi}(t_j, z_j)$ of arguments l ∈ ℕ, t1, t2… ∈ T, z1, z2 ∈ ℝd the covaristic of the measure-valued random function (MVRF)ψ(۔, ۔). A general limit theorem for MVRF's interms of covaristics is proved and applied to functions of thekind ψn(t, B) = µ{x : ξn(t, x) ∈ B}, where μ is anonrandom finite measure and, for each n, ξn is atime-dependent random field.


In this paper we present a stability criterion for processor-sharing queues, in which the throughput may depend on the number of customers in the system (such as in the case of interferences between users). Such a system is represented by a point measure-valued stochastic recursion keeping track of the remaining processing times of the customers.

In 1991 Perkins [7] showed that the normalized critical binary branching process is a time inhomogeneous Fleming-Viot process. In the present paper we extend this result to jump-type branching processes and we show that the normalized jump-type branching processes are in a new class of probability measure-valued processes which will be called ‘jump-type Fleming-Viot processes’. Furthermore we also show that by using these processes it is possible to introduce another new class of measure-valued processes which are obtained by the combination of jump-type branching processes and Fleming-Viot processes.

It is shown that the critical two-level (2, d, 1, 1)-superprocess is persistent in dimensions d greater than 4. This complements the extinction result of Wu (1994) and implies that the critical dimension is 4.

A measure-valued diffusion approximation to a two-level branching structure was introduced in Dawson and Hochberg (1991) where it was shown that conditioned on non-extinction at time t, and appropriately rescaled, the process converges as t → ∞to a non-trivial limiting distribution. Here we discuss a different approach to conditioning on non-extinction (popular in one-level branching) and relate the two limiting distributions.

In this paper we introduce a multilevel birth-death particle system and consider its diffusion approximation which can be characterized as a M([R+)-valued process. The tightness of rescaled processes is proved and we show that the limiting M(R+)-valued process is the unique solution of the M([R+)-valued martingale problem for the limiting generator. We also study the moment structures of the limiting diffusion process.

We consider a dynamic multilevel population or information system. At each level individuals or information units undergo a Galton–Watson-type branching process in which they can be replicated or removed. In addition, a collection of individuals or information units at a given level constitutes an information unit at the next higher level. Each collection of units also undergoes a Galton–Watson branching process, either dying or replicating. In this paper, we represent this multilevel branching model as a measure-valued stochastic process, study its moment structure, identify the limiting continuous-state approximation and analyse the long-time behavior in both non-critical and critical cases. For example, we obtain an asymptotic expression for the extinction probability for the total population mass process and an analogue of Yaglom's conditioned limit theorem in the critical case.