CLT for supercritical branching processes with heavy-tailed branching law

Consider a branching system with particles moving according to an Ornstein-Uhlenbeck process with drift $\mu>0$ and branching according to a law in the domain of attraction of the $(1+\beta)$-stable distribution. The mean of the branching law is strictly larger than $1$ implying that the system is supercritical and the total number of particles grows exponentially at some rate $\lambda>0$. It is known that the system obeys a law of large numbers. In the paper we study its rate of convergence. We discover an interesting interplay between the branching rate $\lambda$ and the drift parameter $\mu$. There are three regimes of the second order behavior: $\cdot$ small branching, $\lambda <(1+1/\beta) \mu$, then the speed of convergence is the same as in the stable central limit theorem but the limit is affected by the dependence between particles. $\cdot$ critical branching, $\lambda =(1+1/\beta) \mu$, then the dependence becomes strong enough to make the rate of convergence slightly smaller, yet the qualitative behaviour still resembles the stable central limit theorem $\cdot$ large branching, $\lambda > (1+1/\beta) \mu$, then the dependence manifests much more profoundly, the rate of convergence is substantially smaller and strangely the limit holds a.s.