Extreme Events of Markov Chains

TL;DR

This paper develops advanced methods to analyze the extremal behavior of Markov chains, especially for asymptotically independent processes, extending existing models to better capture the full evolution of extreme events.

Contribution

It introduces a generalized limiting framework that encompasses a wider class of asymptotically independent Markov chains, extending the Heffernan-Tawn normalization scheme.

Findings

New normalization schemes for asymptotically independent chains

Broader class of Markov processes analyzed

Enhanced understanding of extreme event evolution

Abstract

The extremal behaviour of a Markov chain is typically characterized by its tail chain. For asymptotically dependent Markov chains existing formulations fail to capture the full evolution of the extreme event when the chain moves out of the extreme tail region and for asymptotically independent chains recent results fail to cover well-known asymptotically independent processes such as Markov processes with a Gaussian copula between consecutive values. We use more sophisticated limiting mechanisms that cover a broader class of asymptotically independent processes than current methods, including an extension of the canonical Heffernan-Tawn normalization scheme, and reveal features which existing methods reduce to a degenerate form associated with non-extreme states.