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Limit theorems for supercritical remaining-lifetime age-structured branching processes

Ziling Cheng

Abstract

We study supercritical age-structured branching models starting from a single particle with a random lifetime, where the reproduction law depends on the remaining lifetime of the parent. The lifespan of an individual is decided at its birth and its remaining lifetime decreases at the unit speed. A necessary and sufficient condition is provided for the convergence of the Malthusian normalized random measures. The Malthusian type limit theory in a functional form can be strengthened to hold with probability one under some ``$L\log L$'' conditions. We further prove a central limit theory with a random normalization factor.

Limit theorems for supercritical remaining-lifetime age-structured branching processes

Abstract

We study supercritical age-structured branching models starting from a single particle with a random lifetime, where the reproduction law depends on the remaining lifetime of the parent. The lifespan of an individual is decided at its birth and its remaining lifetime decreases at the unit speed. A necessary and sufficient condition is provided for the convergence of the Malthusian normalized random measures. The Malthusian type limit theory in a functional form can be strengthened to hold with probability one under some ``'' conditions. We further prove a central limit theory with a random normalization factor.
Paper Structure (6 sections, 13 theorems, 205 equations)

This paper contains 6 sections, 13 theorems, 205 equations.

Table of Contents

  1. Introduction
  2. An age-structured branching process
  3. Existence of the non-degenerate limit
  4. Convergence of the Malthusian normalized random measures
  5. Almost sure convergence
  6. A central limit theorem

Key Result

Theorem 2.4

Let $q(t)=\mathbf{P}[X_t (\infty)=0]$ and $q=\mathbf{P}[\lim_{n\rightarrow\infty}\hat{\xi}_{n}=0]$. Then $q(t) \uparrow q \in(0,1]$, as $t \rightarrow \infty$. In particular, $q=1$ if and only if $\mathbf{E}[N(\infty)]\le 1$.

Theorems & Definitions (43)

  • Proof
  • Theorem 2.4
  • Proof
  • Proof
  • Proof
  • Proof
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  • Theorem 3.7
  • ...and 33 more