Convergence of a Critical Multitype Branching Particle System with Mixed Finite and Infinite Mean Lifetimes
Abstract
We study a critical multitype branching particle system in ℝN with finite type space 𝒦=1,…,K. Particles of type i move according to a symmetric αi-stable process and reproduce according to a critical offspring law with irreducible stochastic mean matrix. All lifetime distributions are non-arithmetic. The type-$1$ lifetime distribution has infinite mean and a regularly varying tail 1−F1(t)∼ c1t−γ, 0<γ<1, whereas the lifetime distributions of types 2,…,K satisfy the polynomial upper-tail bounds 1−Fi(t)≤ C t−ηi, i=2,…,K, ηi>1, η:=min2≤ i≤ Kηi. The offspring mechanism satisfies a (1+β)-stable-domain regular-variation condition, with β∈(0,1]. The initial population is a Poisson random measure with intensity Λ=∑i=1K ai λ⊗δi, ai≥0, and we set A:=∑i=1K ai. Under the space--lifetime condition ρ:=(η−1)∧N/α1 > γ/β, we prove that the system converges to a Poisson random measure with intensity Aλ⊗δ1. Thus, although the initial population may assign positive spatial intensity to every type, the limiting population is supported entirely on the infinite-mean type while preserving the aggregate initial spatial intensity Aλ.
Related
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BibTeX
@misc{j2026convergence,
title = {{Convergence of a Critical Multitype Branching Particle System with Mixed Finite and Infinite Mean Lifetimes}},
author = {Ramírez-González J. H. and Prates Machado Fabio},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.11511},
eprint = {2606.11511},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.11511},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Convergence of a Critical Multitype Branching Particle System with Mixed Finite and Infinite Mean Lifetimes
AU - Ramírez-González J. H.
AU - Prates Machado Fabio
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2606.11511
UR - https://arxiv.org/abs/2606.11511
AB - We study a critical multitype branching particle system in ℝ^N with finite type space 𝒦=1,…,K. Particles of type i move according to a symmetric α_i-stable process and reproduce according to a critical offspring law with irreducible stochastic mean matrix. All lifetime distributions are non-arithmetic. The type-$1$ lifetime distribution has infinite mean and a regularly varying tail 1−F_1(t)∼ c_1t^−γ, 0<γ<1, whereas the lifetime distributions of types 2,…,K satisfy the polynomial upper-tail bounds 1−F_i(t)≤ C t^−η_i, i=2,…,K, η_i>1, η:=min_2≤ i≤ Kη_i. The offspring mechanism satisfies a (1+β)-stable-domain regular-variation condition, with β∈(0,1]. The initial population is a Poisson random measure with intensity Λ=∑_i=1^K a_i λ⊗δ_i, a_i≥0, and we set A:=∑_i=1^K a_i. Under the space--lifetime condition ρ:=(η−1)∧N/α_1 > γ/β, we prove that the system converges to a Poisson random measure with intensity Aλ⊗δ_1. Thus, although the initial population may assign positive spatial intensity to every type, the limiting population is supported entirely on the infinite-mean type while preserving the aggregate initial spatial intensity Aλ.
ER -