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

Olivier Wintenberger (1, 2) , Thomas Mikosch (3)

1 LFA - Laboratoire de Finance Assurance
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 UCPH - University of Copenhagen = Københavns Universitet

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.

Mots clés: Markov processes; regular variation; central limit theorem; large deviation principle; GARCH; AMS 2000 subject classifications: Primary 60J05; Secondary 60F10 60F05 60G70 Markov processes regular variation central limit theorem large deviation principle GARCH; AMS 2000 subject classifications: Primary 60J05; Secondary 60F10; 60F05; 60G70 Markov processes

Domaines: Probabilités [math.PR]; Statistiques [math.ST]; Théorie [stat.TH]

Olivier Wintenberger, Thomas Mikosch. The cluster index of regularly varying sequences with applications to limit theory for functions of multivariate Markov chains. Probability Theory and Related Fields, 2013, 159 (1-2), pp.157-196. ⟨10.1007/s00440-013-0504-1⟩. ⟨hal-00806306⟩