Marcinkiewicz--Zygmund-type SLLN for mixed moving average processes
Danijel Grahovac, Péter Kevei, Dominik Mihalčić
TL;DR
We address the problem of establishing a Marcinkiewicz–Zygmund-type SLLN for the time-integrated MMA process $X^*(t)$ by introducing a main tail-moment condition on the driving Lévy measure and kernel: $\int_V \int_{\mathbb{R}} |z|^{\gamma} f_1(x)^{\gamma} \mathbf{1}(|z| f_1(x)>1) \lambda(dz) \pi(dx) < \infty$ for $\gamma\in(0,2]$. The approach combines a decomposition of the Lévy basis, stochastic Fubini, and truncation arguments to treat both integrable and non-integrable small jumps, as well as Gaussian limits; it yields $\frac{X^*(t) - \mathbf{1}(\gamma\ge1)\mathbb{E}X^*(t)}{t^{1/\gamma}} \to 0$ a.s. (and a LIL when $\gamma=2$) and explicit rate bounds depending on tail indices $\alpha$, $\beta$, and $\eta$. The results apply to broad MMA models, including supOU, trawl, and supfOU, with explicit forms for $f_1$ and $f_2$ and verifiable assumptions, thereby providing sharp, model-wide insights into almost-sure growth rates under strong dependence. Overall, the paper delivers a unified framework for almost-sure asymptotics of time-integrated MMA processes and clarifies how dependence and marginal-tail features govern growth rates in concrete stationary infinitely divisible settings.
Abstract
The Marcinkiewicz--Zygmund theorem is a fundamental result in probability theory that establishes rates of convergence in the strong law of large numbers (SLLN). Although numerous extensions have been developed for dependent sequences, many classes of processes, particularly those exhibiting strong dependence, remain unexplored. In this paper, we present a Marcinkiewicz--Zygmund-type SLLN for a class of mixed moving average processes, which form a large and flexible class of stationary infinitely divisible processes. In contrast to the classical case, where moments determine the asymptotic behavior, the present setting additionally involves key objects that characterize both dependence and marginal distributions.