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 $\mathbb{R}^N$ with finite type space $\mathcal{K}={1,\dots,K}$. Particles of type $i$ move according to a symmetric $\alpha_i$-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 $$ 1-F_1(t)\sim c_1t^{-\gamma}, \qquad 0<\gamma<1, $$ whereas the lifetime distributions of types $2,\dots,K$ satisfy the polynomial upper-tail bounds $$ 1-F_i(t)\le C t^{-\eta_i}, \qquad i=2,\dots,K, \qquad \eta_i>1, \qquad \eta:=\min_{2\le i\le K}\eta_i. $$ The offspring mechanism satisfies a $(1+\beta)$-stable-domain regular-variation condition, with $\beta\in(0,1]$. The initial population is a Poisson random measure with intensity $$ \Lambda=\sum_{i=1}^K a_i\,\lambda\otimes\delta_i, \qquad a_i\ge0, $$ and we set $A:=\sum_{i=1}^K a_i$. Under the space--lifetime condition $$ \rho:=\left(\eta-1\right)\wedge\frac{N}{\alpha_1} > \frac{\gamma}{\beta}, $$ we prove that the system converges to a Poisson random measure with intensity $A\lambda\otimes\delta_1$. 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 $A\lambda$.