Convergence of a Critical Multitype Bellman--Harris Process with Finite-Mean Lifetimes

2024/10/22 by Ekaterina T. Kolkovska, Kolkovska, E. T., José Alfredo López-Mimbela +3
Economics, Econometrics and Finance · Mathematics · Medicine · #60J80 #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2410.17369

openalex publication_date 2024/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study a critical multitype Bellman--Harris branching particle system in \(\mathbb R^N\) with a finite type space \(\mathbf K=\1,…,K\\). Particles of type \(I\) move according to a symmetric \(α_i\)-stable process, have non-arithmetic lifetimes with finite mean, and reproduce according to a critical offspring law whose mean matrix is irreducible and stochastic. The branching mechanism is assumed to be in the domain of attraction of a \((1+β)\)-stable law, with \(β∈(0,1]\). We prove that the particle system converges, as \(t→∞\), to a limiting random measure which is nonzero. We also show that the limiting population preserves the initial intensity measure. Thus, the system persists with full intensity. These results complement the local extinction results of Kevei and Lopez-Mimbela (2011) for critical multitype Bellman--Harris systems.

Related

• Critical Multitype Branching Systems: Extinction Results
• A two-type Bellman--Harris process initiated by a large number of particles
• Reduced critical Bellman-Harris branching processes for small populations
• Critical Galton-Watson branching processes with countably infinitely many types and infinite second moments
• Critical multitype branching: a functional limit theorem and ancestral processes