Ergodic Properties of Max-Infinitely Divisible Processes

TL;DR

This paper establishes that a stationary max-infinitely divisible process is mixing if and only if its dependence function diminishes to zero, providing criteria for ergodicity in such processes.

Contribution

It introduces necessary and sufficient conditions for ergodicity of stationary max-infinitely divisible processes based on their dependence functions.

Findings

Dependence function converging to zero implies mixing.

Cesaro summability of the dependence function implies ergodicity.

Criteria are applied to specific classes of max-infinitely divisible processes.

Abstract

We prove that a stationary max--infinitely divisible process is mixing (ergodic) iff its dependence function converges to 0 (is Cesaro summable to 0). These criteria are applied to some classes of max--infinitely divisible processes.