Markov tail chains

TL;DR

This paper extends the theory of tail chains from univariate to multivariate Markov chains with regularly varying distributions, analyzing their forward and backward tail processes and their applications in extreme value analysis.

Contribution

It generalizes the tail chain concept to multivariate Markov chains and explores the relationship between forward and backward tail processes, including non-Markovian cases.

Findings

Backward tail chain is Markovian if the forward tail chain is Markovian.

Provides asymptotic distributions for past and future conditioned on extremes.

Characterizes limiting processes for multivariate Markov chains with heavy tails.

Abstract

The extremes of a univariate Markov chain with regulary varying stationary marginal distribution and asymptotically linear behavior are known to exhibit a multiplicative random walk structure called the tail chain. In this paper, we extend this fact to Markov chains with multivariate regularly varying marginal distribution in R^d. We analyze both the forward and the backward tail process and show that they mutually determine each other through a kind of adjoint relation. In a broader setting, it will be seen that even for non-Markovian underlying processes a Markovian forward tail chain always implies that the backward tail chain is Markovian as well. We analyze the resulting class of limiting processes in detail. Applications of the theory yield the asymptotic distribution of both the past and the future of univariate and multivariate stochastic difference equations conditioned on an extreme event.