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

We study a critical multitype Bellman--Harris branching particle system in \(\mathbb R^N\) with a finite type space \(\mathbf K=\{1,\dots,K\}\). Particles of type \(I\) move according to a symmetric \(\alpha_i\)-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 \((1+\beta)\)-stable law, with \(\beta\in(0,1]\). We prove that the particle system converges, as \(t\to\infty\), 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.