The cluster index of regularly varying sequences with applications to limit theory for functions of multivariate Markov chains

TL;DR

This paper introduces the cluster index for multivariate regularly varying sequences, linking it to the spectral tail process, and demonstrates its importance in limit theorems and large deviation results for sums of functions of Markov chains.

Contribution

It defines the cluster index for multivariate sequences and applies it to characterize stable limits and large deviations in Markov chain function sums.

Findings

Cluster index characterizes tail behavior in multivariate sequences.

Stable limit distributions are identified using the cluster index.

Precise large deviation results are derived for Markov chain functions.

Abstract

We introduce the cluster index of a multivariate regularly varying stationary sequence and characterize the index in terms of the spectral tail process. This index plays a major role in limit theory for partial sums of regularly varying sequences. We illustrate the use of the cluster index by characterizing infinite variance stable limit distributions and precise large deviation results for sums of multivariate functions acting on a stationary Markov chain under a drift condition.