Symmetric Stable Processes on Amenable Groups

Nachi Avraham-Re'em

Published 2022-05-09Version 1

We show that if $G$ is a countable amenable group, then every stationary non-Gaussian symmetric stable ($S\alpha S$) process indexed by $G$ is ergodic if and only if it is weakly-mixing, and it is ergodic if and only if its Rosi\'nski minimal spectral representation is purely null. This extends the results of Samorodnitsky for $\mathbb{Z}$ and Wang, Roy & Stoev for $\mathbb{Z}^d$, and answers a question of P. Roy about discrete nilpotent groups to the extent of all countable amenable groups.