Long-time behaviour of Galton-Watson systems with circular mechanism

Junping Li; Mixuan Hou

Long-time behaviour of Galton-Watson systems with circular mechanism

Junping Li, Mixuan Hou

Abstract

This paper concentrates on the limit behavior of discrete-time branching process with circular mechanism. Three types of limit behaviour of discrete-time branching process with circular mechanism are given explicitly under various moment conditions on branching rates. It is proved that the rate of the first one is geometric, while the other two are supergeometric.

Long-time behaviour of Galton-Watson systems with circular mechanism

Abstract

Paper Structure (17 theorems, 52 equations)

This paper contains 17 theorems, 52 equations.

Key Result

Lemma 2.1

For any $\textbf{a}=\{a_k\}_{k=0}^{\infty}\in \mathscr{P}$, $f(\textbf{a};s)$ is a convex increasing function on $[0,1]$. If $m_a\leq 1$ then $f(\textbf{a};s)>s$ for all $s \in [0,1)$ and $f(\textbf{a};s)=s$ has exactly one root $1$ on $[0,1]$. Furthermore, if $m_a<1$ then $1$ is a simple root whil

Theorems & Definitions (20)

Definition 1.1
Lemma 2.1
Lemma 2.2
Definition 2.2
Theorem 2.1
Remark 2.1
Lemma 2.3
Theorem 2.2
Theorem 3.1
Theorem 3.2
...and 10 more