Geometric ergodicity for some space-time max-stable Markov chains

TL;DR

This paper proves geometric ergodicity of certain space-time max-stable Markov chains used for modeling spatial extremes, overcoming challenges posed by the non-locally compact state space.

Contribution

It establishes geometric ergodicity for specific max-stable Markov chains on a non-locally compact space using advanced probabilistic techniques.

Findings

Markov chains are geometrically ergodic

State space is Polish but not locally compact

Classical methods are inapplicable due to non-local compactness

Abstract

Max-stable processes are central models for spatial extremes. In this paper, we focus on some space-time max-stable models introduced in Embrechts et al. (2016). The processes considered induce discrete-time Markov chains taking values in the space of continuous functions from the unit sphere of $\mathbb{R}^3$ to $(0, \infty)$. We show that these Markov chains are geometrically ergodic. An interesting feature lies in the fact that the state space is not locally compact, making the classical methodology inapplicable. Instead, we use the fact that the state space is Polish and apply results presented in Hairer (2010).