A multilevel branching model

Abstract

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.

Limiting behaviour of two-level measure-branching

“…There is now a substantial literature on (measure-valued) diffusion approximations to one-level critical (or near-critical) branching particle systems. Dawson and Hochberg (1991) introduce a measure-valued diffusion approximation to a two-level branching process. Such multilevel structures have been used, for example, to describe the replication, updating and transfer of digitized data sets as they pass through information networks (see Dawson and Hochberg for applications to other areas such as population biology).…”

“…Set 6Before performing the rescaling in (4), let us give another expression for YrljJ. By the Feynman-Kac formula, YrljJ satisfies it suffices to study Yr 0 atljJ(x) for which we use the following observation of Dawson and Hochberg (1991)…”

“…If ep E ct(\R +), then W~</>(x)~x and Us</> is uniformly bounded. Thus l./\'ep~.ep E C(j(~+) and so we may use the Poisson cluster representation for continuous-state branching (see Dawson and Hochberg (1991), Equation 6.16). We see that where p,(x) =~exp ( -~) .…”

A Multilevel birth-death particle system and its continuous diffusion

“…In this paper, we introduce a multilevel birth-death particle model, which is a generalization of the multilevel branching model introduced by Dawson and Hochberg [4], and then consider the measure-valued diffusion approximation to the multilevel birth-death particle system. In Section 2, we describe the two-level birth-death particle system as a N(Z+)-valued process.…”

Persistence of a critical super-2 process

“…Super-2 processes (or 2-level superprocesses) have been investigated recently by Dawson, Hochberg, Vinogradov and Wu ([4], [5], [6], [7], [13], [15], [16], [17]). They arise naturally as models for hierarchically structured two-level populations or particle systems in which individuals undergo some spatial motion in (Rd -say, a symmetric stable diffusion of index a, where 0 < a~2 -and branch at random times with some branching parameter PI' where 0 < PI~1.…”