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

Abstract

We introduce the cluster index of a multivariate 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 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. © 2013 Springer-Verlag Berlin Heidelberg.