Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An inhomogeneous controlled branching process

Miguel González, Carmen Minuesa, Manuel Mota, Inés del Puerto, Alfonso Ramos

Abstract

A discrete time branching process where the offspring distribution is generation-dependent, and the number of reproductive individuals is controlled by a random mechanism is considered. This model is a Markov chain but, in general, the transition probabilities are non-stationary. Under not too restrictive hypotheses, this model presents the classical duality of branching processes: either becomes extinct almost surely or grows to infinity. Sufficient conditions for the almost sure extinction and for a positive probability of indefinite growth are provided. Finally rates of growth of the process provided the non-extinction are studied.

An inhomogeneous controlled branching process

Abstract

A discrete time branching process where the offspring distribution is generation-dependent, and the number of reproductive individuals is controlled by a random mechanism is considered. This model is a Markov chain but, in general, the transition probabilities are non-stationary. Under not too restrictive hypotheses, this model presents the classical duality of branching processes: either becomes extinct almost surely or grows to infinity. Sufficient conditions for the almost sure extinction and for a positive probability of indefinite growth are provided. Finally rates of growth of the process provided the non-extinction are studied.
Paper Structure (4 sections, 8 theorems, 49 equations)

This paper contains 4 sections, 8 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. The extinction problem
  3. Asymptotic behaviour
  4. Proofs

Key Result

Theorem 1

Let $\{Z_n\}_{n\ge0}$ be a CPVE satisfying: Then

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 1
  • Theorem 4
  • Theorem 5
  • Proposition 6
  • Theorem 7
  • Lemma 8