Controlled branching processes with continuous time

Abstract

A class of controlled branching processes with continuous time is introduced and some limiting distributions are obtained in the critical case. An extension of this class as regenerative controlled branching processes with continuous time is proposed and some asymptotic properties are considered.

Simulation study on the offspring mean estimators in discrete time branching processes with migration

“…There are numerous sources providing the definition of a controlled branching process introduced by Sevastyanov in [9]. For the purpose of this article, we will give the formulation presented in [4].…”

“…A special case of the MCB process is the one with random migration [4]. Let X = {X n (i), n = 0, 1, .…”

“…A simpler variant of model (3), introduced above, is presented in [3], where emigration and immigration take place before the reproduction phase.…”

“…For the simulation of (5), we once again assume that p, q, and r represent the probabilities of emigration, no migration, and immigration, respectively. This is analog of Gawton-Watson process with migration defined in [3], in which the control function is a random variable, depending on p, q and r. Here p = 0.03, q = 0.2046, r = 0.7654 and the offspring mean is m ≈ 1.0993, starting from Z 0 = 1. The results are presented in Table 4.…”

Branching processes in varying environment with generation-dependent immigration

“…In particular, special attention has been paid to the problem of modelling migratory movements. Within the class of discrete time models, it is worthwhile to mention the branching process with immigration, the branching process with time-dependent immigration, the branching process with immigration only after empty generation, branching processes with migration or the branching process with immigration in varying or random environment (for their description, see the pioneer works [11,12], [4], [22], [21] and [18], respectively, or [6] for a review of them) and the age-dependent branching process with immigration or its generalization, the Sevast'yanov branching process with immigration (see [15] and [28], respectively, for their description), in continuous time among others.…”