Marcinkiewicz--Zygmund-type SLLN for mixed moving average processes

TL;DR

This paper extends the Marcinkiewicz--Zygmund strong law of large numbers to mixed moving average processes, a broad class of dependent, stationary infinitely divisible processes, highlighting the role of dependence and distributional characteristics.

Contribution

It introduces a Marcinkiewicz--Zygmund-type SLLN for mixed moving average processes, addressing dependence and distributional factors not covered in classical results.

Findings

Established a strong law of large numbers for mixed moving average processes.

Identified key objects characterizing dependence and marginal distributions.

Extended classical results to a broader class of dependent processes.

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.