Law of the iterated logarithm for supercritical non-local spatial branching processes

TL;DR

This paper establishes iterated logarithm laws for martingales associated with supercritical non-local branching processes, revealing three regimes with different scaling limits and extending classical results to more general non-local and non-symmetric settings.

Contribution

It introduces new law of the iterated logarithm results for supercritical non-local branching processes, generalizing previous work to include non-local mechanisms and non-symmetric spatial motions.

Findings

Identified three regimes with distinct scaling factors and limits.

Established LIL for linear functionals of the process involving eigenfunctions.

Extended classical results to non-local, non-symmetric branching processes.

Abstract

Suppose that $X=(X_{t})_{t\ge 0}$ 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 $\langle \mathrm{Re}(f),X_{t}\rangle$ where $f$ is a sum of finite terms of eigenfunctions and $\mathrm{Re}(f)$ 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: 2505.12691] to allow for non-local branching mechanism and non-symmetric spatial motion. For superprocesses, as far as we know, our results are new.