Convergence of a Critical Multitype Bellman--Harris Process with One Infinite-Mean Lifetime

Abstract

We study a critical multitype Bellman--Harris branching particle system in $\mathbb R^N$ with a finite type space $\mathbb K=\{1,\dots,K\}$. Particles of type $i$ move according to a symmetric $α_i$-stable process and reproduce according to a critical offspring law whose mean matrix is irreducible and stochastic. The lifetime distribution of type $1$ is assumed to have infinite mean with regularly varying tail $$ 1-F_1(t)\sim c_1t^{-γ},\, 0<γ<1, $$ whereas the remaining lifetime distributions satisfy polynomial upper-tail bounds $$ \overline F_i(t)\le C t^{-η_i},\, i=2,\dots,K, \, η_i>1, \, η:=\min_{2\le i\le K}η_i. $$ The branching mechanism is assumed to be in the domain of attraction of a $(1+β)$-stable law, with $β\in(0,1]$. Under the space--lifetime condition $$ ρ:=\left(η-1\right)\wedge\frac{N}{α_1} > \fracγβ, $$ and a local increment condition on the heavy lifetime distribution, we prove convergence of the system to a Poisson random measure concentrated on the infinite-mean type.