Embeddability of Discrete Time Simple Branching Processes into Continuous Time Branching Processes

Simple Branching Processes

Consider a particle that can replicate itself instantaneously at random times. In general there is a distribution of times from which the particle draws in order to initiate replication. For simplicity, we assume a Poisson process, and a replication rate of λ. Consider the total integer number of particles N(t) at any time t. Let us now consider one realization of the continuous-time branching ...

An Extended Class of Time-continuous Branching Processes

This paper is devoted to studying an extended class of time-continuous branching processes, motivated by the study of stochastic control theory and interacting particle systems. The uniqueness, extinction, recurrence and positive recurrence criteria for the processes are presented. The main new point in our proofs is the use of several different comparison methods. The resulting picture shows t...

Limiting Distributions for Critical Multitype Branching Processes with Discrete Time(')

General branching processes in discrete time as random trees

The simple Galton–Watson process describes populations where individuals live one season and are then replaced by a random number of children. It can also be viewed as a way of generating random trees, each vertex being an individual of the family tree. This viewpoint has led to new insights and a revival of classical theory. We show how a similar reinterpretation can shed new light on the more...

Multi-type branching processes with time-dependent branching rates

Under mild non-degeneracy assumptions on branching rates in each generation, we provide a criterion for almost-sure extinction of a multi-type branching process with time-dependent branching rates. We also provide a criterion for the total number of particles (conditioned on survival and divided by the expectation of the resulting random variable) to approach an exponential random variable as t...