2025/08/18 by Haojie Hou, Ting Yang, Hou, Haojie +1
Economics, Econometrics and Finance · Mathematics · #Context (archaeology) #Iterated function #Iterated logarithm #Law of the iterated logarithm #Logarithm #Markov chain #Markov process #Semigroup #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #Supercritical fluid #advanced mathematical theories #math.PR
paper · pdf · doi:10.48550/arxiv.2508.12772
openalex publication_date 2025/08/18 · openalex created_date 2025/10/18 · openalex updated_date 2026/08/05
Suppose that X=(Xt)t≥ 0 is either a general supercritical non-local branching Markov process, or a general supercritical non-local superprocess, on a Luzin space. Here, by ``supercritical" we mean that the mean semigroup of X exhibits a Perron-Frobenius type behaviour with a positive principal eigenvalue. In this paper, we study the almost sure behaviour of a family of martingales naturally associated with the real or complex-valued eigenpairs of the mean semigroup. Under a fourth-moment condition, we establish limit theorems of the iterated logarithm type for these martingales. In particular, we discover three regimes, each resulting in different scaling factors and limits. Furthermore, we obtain a law of the iterated logarithm for the linear functional ⟨ Re(f),Xt⟩ where f is a sum of finite terms of eigenfunctions and Re(f) denotes its real part. In the context of branching Markov processes, our results improve on existing literature by complementing the known results for multitype branching processes in Asmussen [Trans. Amer. Math. Soc. 231 (1) (1977) 233--248] and generalizing the recent work of Hou, Ren and Song [arXiv: 2505.12691v1, arXiv: 2505.12691v2] to allow for non-local branching mechanisms. For superprocesses, as far as we know, our results are new.