2018/03/14 by Rafał Marks, Piotr Miłoś, Marks, Rafał +1
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #60J80 #FOS: Mathematics #Primary 60F05 #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #secondary 60G20
paper · pdf · doi:10.48550/arxiv.1803.05491
openalex publication_date 2018/03/14 · openalex created_date 2018/03/29 · openalex updated_date 2026/07/28
Consider a branching system with particles moving according to an Ornstein-Uhlenbeck process with drift μ>0 and branching according to a law in the domain of attraction of the (1+β)-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 λ>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 λ and the drift parameter μ. There are three regimes of the second order behavior: ⋅ small branching, λ (1+1/β) μ, then the dependence manifests much more profoundly, the rate of convergence is substantially smaller and strangely the limit holds a.s.