A functional ergodic theorem for the occupation time process of a branching system

We prove a functional ergodic theorem for the occupation time process of a (d,α,β)(d,α,β)-branching particle system (particles moving in RdRd according to a spherically symmetric αα-stable Lévy process, (1+β)(1+β)-branching, 0<β≤10<β≤1, Poisson initial state with Lebesgue intensity measure), in the critical case d=αβ.