# Mixing properties of multivariate infinitely divisible random fields

## TL;DR

This paper investigates the mixing properties of multivariate infinitely divisible stationary random fields, providing conditions for mixing, demonstrating that mixed moving average fields are mixing, and establishing the equivalence of ergodicity and weak mixing.

## Contribution

It offers new necessary and sufficient conditions for mixing in multivariate ID random fields and proves that mixed moving average fields are mixing, also linking ergodicity with weak mixing.

## Key findings

- Derived spectral conditions for mixing
- Proved mixed moving average fields are mixing
- Established ergodicity and weak mixing equivalence

## Abstract

In this work we present different results concerning mixing properties of multivariate infinitely divisible (ID) stationary random fields. First, we derive some necessary and sufficient conditions for mixing of stationary ID multivariate random fields in terms of their spectral representation. Second, we prove that (linear combinations of independent) mixed moving average fields are mixing. Further, using a simple modification of the proofs of our results we are able to obtain weak mixing versions of our results. Finally, we prove the equivalence of ergodicity and weak mixing for multivariate ID stationary random fields.