Clustering of Markov chain exceedances

TL;DR

This paper studies how extreme events in Markov chains cluster together by analyzing the tail chain and showing convergence to a cluster Poisson process, offering a new regenerative cycle approach.

Contribution

It introduces a novel method using regenerative cycles to analyze extremal dependence in Markov chains, differing from traditional blocking techniques.

Findings

Convergence of normalized extreme observations to a cluster Poisson process.

Decomposition of sample paths into i.i.d. regenerative cycles.

General conditions under which the tail chain models extremal dependence.

Abstract

The tail chain of a Markov chain can be used to model the dependence between extreme observations. For a positive recurrent Markov chain, the tail chain aids in describing the limit of a sequence of point processes $\{N_n,n\geq1\}$, consisting of normalized observations plotted against scaled time points. Under fairly general conditions on extremal behaviour, $\{N_n\}$ converges to a cluster Poisson process. Our technique decomposes the sample path of the chain into i.i.d. regenerative cycles rather than using blocking argument typically employed in the context of stationarity with mixing.