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Theta-positive branching in varying environment

Serik Sagitov, Alexey Lindo, Yerakhmet Zhumayev

TL;DR

This work analyzes theta-positive branching processes in continuous time under a time-varying environment, deriving explicit limit theorems that classify behavior into five regimes: supercritical, asymptotically degenerate, critical, strictly subcritical, and loosely subcritical. Central to the results is the explicit generating function $E(s^{Z_t})=1-(B_{t,\theta}+\mu_t^{-\theta}(1-s)^{-\theta})^{-1/\theta}$ and the key quantities $\mu_t$, $\Lambda_t$, $V_{t,\theta}$, and $V_{\theta}$, which determine extinction probabilities, growth rates, and limiting distributions (including a nontrivial limit $W$ in the supercritical case and $Z_\infty$ in the degenerate case). The paper provides six limit theorems, detailed asymptotics for $P(Z_t>0)$, $m_t$, and the Laplace or generating function limits, and a suite of illustrative examples showing regime transitions under various time-varying mean offspring $a_t$ and environment schedules. By treating $0<\theta\le1$ and allowing infinite-variance offspring, the results extend classical homogeneous models to richly varying environments with explicit, tractable limit laws, informing applications in population dynamics and related stochastic processes.

Abstract

Branching processes in a varying environment encompass a wide range of stochastic demographic models, and their complete understanding in terms of limit behaviour poses a formidable research challenge. In this paper, we conduct a thorough investigation of such processes within a continuous-time framework, assuming that the reproduction law of individuals adheres to a specific parametric form for the probability generating function. Our six clear-cut limit theorems support the notion of recognizing five distinct asymptotical regimes for branching in varying environments: supercritical, asymptotically degenerate, critical, strictly subcritical, and loosely subcritical.

Theta-positive branching in varying environment

TL;DR

This work analyzes theta-positive branching processes in continuous time under a time-varying environment, deriving explicit limit theorems that classify behavior into five regimes: supercritical, asymptotically degenerate, critical, strictly subcritical, and loosely subcritical. Central to the results is the explicit generating function and the key quantities , , , and , which determine extinction probabilities, growth rates, and limiting distributions (including a nontrivial limit in the supercritical case and in the degenerate case). The paper provides six limit theorems, detailed asymptotics for , , and the Laplace or generating function limits, and a suite of illustrative examples showing regime transitions under various time-varying mean offspring and environment schedules. By treating and allowing infinite-variance offspring, the results extend classical homogeneous models to richly varying environments with explicit, tractable limit laws, informing applications in population dynamics and related stochastic processes.

Abstract

Branching processes in a varying environment encompass a wide range of stochastic demographic models, and their complete understanding in terms of limit behaviour poses a formidable research challenge. In this paper, we conduct a thorough investigation of such processes within a continuous-time framework, assuming that the reproduction law of individuals adheres to a specific parametric form for the probability generating function. Our six clear-cut limit theorems support the notion of recognizing five distinct asymptotical regimes for branching in varying environments: supercritical, asymptotically degenerate, critical, strictly subcritical, and loosely subcritical.
Paper Structure (14 sections, 9 theorems, 73 equations, 1 figure)

This paper contains 14 sections, 9 theorems, 73 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Examples
  4. An example with $a_t\searrow1$
  5. An example with $a_t\nearrow1$
  6. An example with $a_t\searrow0$
  7. A loosely subcritical case
  8. Proofs
  9. Derivation of \ref{['MA']}, \ref{['thetaF']}, and \ref{['thetaB']}
  10. Proof of Theorem \ref{['Thm0']}
  11. Proof of Theorem \ref{['Thm1']}
  12. Proof of Theorem \ref{['Thm2']}
  13. Proof of Theorem \ref{['Thm3']}
  14. Proof of Theorem \ref{['Thm4']} and Theorem \ref{['Thm5']}

Key Result

Theorem 2.1

If $V_\theta<\infty$, then $q<1$, and If $V_\theta=\infty$, then $q=1$ and

Figures (1)

  • Figure 1: On the plot, the horizontal axis gives the time varying $t$ and the vertical axis gives $\mu_t$ for the example of Section \ref{['vast']}.

Theorems & Definitions (9)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Theorem 2.5
  • Corollary 2.6
  • Theorem 2.7
  • Corollary 2.8
  • Theorem 2.9