Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Critical branching processes with immigration: scaling limits of local extinction sets

Aleksandar Mijatović, Benjamin Povar, Gerónimo Uribe Bravo

TL;DR

The paper studies scaling limits for a critical Bienaymé-Galton-Watson process with immigration (BGWI) and its counting local time at zero, establishing convergence to a self-similar continuous-state branching process with immigration (CBI) and its Markovian local time at zero under stable-domain attraction with index $\alpha\in(0,1]$. Using a general invariance-principle framework for local times, the authors reduce the problem to excursion structure and hitting-time convergence, and they prove a new Yaglom limit for infinite-variance BGWIs. The main result is a septuple scaling limit that couples the BGWI, its local time, and associated random walks through a continuous Lamperti transformation, with explicit scaling sequences and a detailed description of limiting objects: a stable Lévy pair $(X,Y)$, their functionals, and the CBI pair $(Z,L)$. The work provides new tools for analyzing local-time functionals in heavy-tailed branching systems and solidifies the connection between pre-limit BGWIs and continuum-time CBI processes, with implications for understanding the zero-set structure and excursion dynamics.

Abstract

We establish the joint scaling limit of a critical Bienaymé-Galton-Watson process with immigration (BGWI) and its (counting) local time at zero to the corresponding self-similar continuous-state branching process with immigration (CBI) and its (Markovian) local time at zero for balanced offspring and immigration laws in stable domains of attraction. Using a general framework for invariance principles of local times~\cite{MR4463082}, the problem reduces to the analysis of the structure of excursions from zero and positive levels, together with the weak convergence of the hitting times of points of the BGWI to those of the CBI. A key step in the proof of our main limit theorem is a novel Yaglom limit for the law at time $t$ of an excursion with lifetime exceeding $t$ of a scaled infinite-variance critical BGWI. Our main result implies a joint septuple scaling limit of BGWI $Z_1$, its local time at $0$, the random walks $X_1$ and $Y_1$ associated to the reproduction and immigration mechanisms, respectively, the counting local time at $0$ of $X_1$, an additive functional of $Z_1$ and $X_1$ evaluated at this functional. In the septuple limit, four different scaling sequences are identified and given explicitly in terms of the offspring generating function (modulo asymptotic inversion), the local extinction probabilities of the BGWI and the tails of return times to zero of $X_1$.

Critical branching processes with immigration: scaling limits of local extinction sets

TL;DR

The paper studies scaling limits for a critical Bienaymé-Galton-Watson process with immigration (BGWI) and its counting local time at zero, establishing convergence to a self-similar continuous-state branching process with immigration (CBI) and its Markovian local time at zero under stable-domain attraction with index . Using a general invariance-principle framework for local times, the authors reduce the problem to excursion structure and hitting-time convergence, and they prove a new Yaglom limit for infinite-variance BGWIs. The main result is a septuple scaling limit that couples the BGWI, its local time, and associated random walks through a continuous Lamperti transformation, with explicit scaling sequences and a detailed description of limiting objects: a stable Lévy pair , their functionals, and the CBI pair . The work provides new tools for analyzing local-time functionals in heavy-tailed branching systems and solidifies the connection between pre-limit BGWIs and continuum-time CBI processes, with implications for understanding the zero-set structure and excursion dynamics.

Abstract

We establish the joint scaling limit of a critical Bienaymé-Galton-Watson process with immigration (BGWI) and its (counting) local time at zero to the corresponding self-similar continuous-state branching process with immigration (CBI) and its (Markovian) local time at zero for balanced offspring and immigration laws in stable domains of attraction. Using a general framework for invariance principles of local times~\cite{MR4463082}, the problem reduces to the analysis of the structure of excursions from zero and positive levels, together with the weak convergence of the hitting times of points of the BGWI to those of the CBI. A key step in the proof of our main limit theorem is a novel Yaglom limit for the law at time of an excursion with lifetime exceeding of a scaled infinite-variance critical BGWI. Our main result implies a joint septuple scaling limit of BGWI , its local time at , the random walks and associated to the reproduction and immigration mechanisms, respectively, the counting local time at of , an additive functional of and evaluated at this functional. In the septuple limit, four different scaling sequences are identified and given explicitly in terms of the offspring generating function (modulo asymptotic inversion), the local extinction probabilities of the BGWI and the tails of return times to zero of .

Paper Structure

This paper contains 15 sections, 16 theorems, 140 equations.

Table of Contents

  1. Introduction and main results
  2. Scaling limits of the critical BGWIs and their counting local times at zero
  3. The septuple limit theorem
  4. Organisation of the remainder of the paper
  5. Proof of Theorem \ref{['theorem_Local_time_conv']}
  6. Convergence of the hitting times of BGW and BGWI: proofs of the Yaglom limit and Theorem \ref{['propositionConvergenceOfPairsdepsdgepsg']}
  7. The local limit theorem for gt(Zn)/t and the proof of the Yaglom limit
  8. Conclusion of the proof of Theorem \ref{['propositionConvergenceOfPairsdepsdgepsg']}
  9. Proof of the weak limit in \ref{['eq:second_weak_limit_in_Thm_5']}
  10. Proof of the weak limit in \ref{['eq:first_weak_limit_in_Thm_5']}
  11. On the tail behaviour of mu and nu and regular variation of a certain scaling sequence
  12. Regular variation of the tails of mu and nu in Assumptions \ref{['assumption_SL']}
  13. Behaviour of mu and nu for alpha equals 1 in Assumptions \ref{['assumption_SL']}
  14. Regular variation of the sequence (an) in Theorem \ref{['septupleLimitTheorem']}
  15. A sequence of BGWIs with a self-similar CBI limit and local times that do not converge to the local time of the CBI

Key Result

Theorem 1

Let Assumption assumption_SL hold with $\delta=\frac{d}{\alpha c}\in (0,1)$. Then, for any $\kappa>0$, the sequence $(c_n)$ given by is regularly varying of index $1-\delta$. Furthermore, there exists $\kappa$ so that the weak convergence holds in the product Skorokhod topology.

Theorems & Definitions (36)

  • Theorem 1
  • Remark
  • Theorem 2: MR4463082
  • Theorem 3: Yaglom limit for BGWI
  • Theorem 4
  • proof : Proof of Theorem \ref{['septupleLimitTheorem']}
  • proof : Proof of Theorem \ref{['theorem_Local_time_conv']} (case $\lim_{ \varepsilon\to0}\mathrm{I}_{d,g}=0$ for $\mathrm{I}_{d,g}$ in \ref{['eq:def_I_II_III']})
  • Lemma 5
  • proof
  • proof : Proof of Theorem \ref{['theorem_Local_time_conv']} (case $\lim_{ \varepsilon\to 0}\lim_{n\to\infty}\mathrm{II}_{d}=0$ for $\mathrm{II}_{d}$ in \ref{['eq:def_I_II_III']})
  • ...and 26 more