The tail empirical process of regularly varying functions of geometrically ergodic Markov chains

TL;DR

This paper studies the tail behavior of regularly varying functions of geometrically ergodic Markov chains, providing conditions for weak convergence of tail sums and estimators, with applications and counterexamples.

Contribution

It introduces practical conditions, including geometric drift, for the weak convergence of tail array sums in Markovian time series models.

Findings

Conditions for weak convergence are established.

Feasible estimators for cluster statistics are proposed.

Counterexample shows different behavior without geometric drift.

Abstract

We consider a stationary regularly varying time series which can be expressedas a function of a geometrically ergodic Markov chain. We obtain practical conditionsfor the weak convergence of the tail array sums and feasible estimators ofcluster statistics. These conditions include the so-called geometric drift or Foster-Lyapunovcondition and can be easily checked for most usual time series models witha Markovian structure. We illustrate these conditions on several models and statisticalapplications. A counterexample is given to show a different limiting behaviorwhen the geometric drift condition is not fulfilled.