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

```@article{Mikosch2013TheCI,
title={The cluster index of regularly varying sequences with applications to limit theory for functions of multivariate Markov chains},
author={Thomas Mikosch and Olivier Wintenberger},
journal={Probability Theory and Related Fields},
year={2013},
volume={159},
pages={157-196}
}```
• Published 2013
• Mathematics
• Probability Theory and Related Fields
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.
The tail empirical process of regularly varying functions of geometrically ergodic Markov chains
• Mathematics
• Stochastic Processes and their Applications
• 2019
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 tailExpand
Estimation of cluster functionals for regularly varying time series: runs estimators
• Mathematics
• 2021
Cluster indices describe extremal behaviour of stationary time series. We consider runs estimators of cluster indices. Using a modern theory of multivariate, regularly varying time series, we obtainExpand
On joint weak convergence of partial sum and maxima processes
For a strictly stationary sequence of random variables we derive functional convergence of the joint partial sum and partial maxima process under joint regular variation with index and weakExpand
Complete convergence theorem for stationary heavy tailed sequences
• Mathematics
• 2015
For a class of stationary regularly varying and weakly dependent time series, we prove the so-called complete convergence result for the corresponding space-time point processes. As an application ofExpand
Estimation of cluster functionals for regularly varying time series: sliding blocks estimators
• Mathematics
• 2020
Cluster indices describe extremal behaviour of stationary time series. We consider their sliding blocks estimators. Using a modern theory of multivariate, regularly varying time series, we obtainExpand
Stable limits for Markov chains via the Principle of Conditioning
• Mathematics
• 2018
Abstract We study limit theorems for partial sums of instantaneous functions of a homogeneous Markov chain on a general state space. The summands are heavy-tailed and the limits are stableExpand
A large deviations approach to limit theory for heavy-tailed time series
• Mathematics
• 2015
In this paper we propagate a large deviations approach for proving limit theory for (generally) multivariate time series with heavy tails. We make this notion precise by introducing regularly varyingExpand
An invariance principle for sums and record times of regularly varying stationary sequences
• Mathematics
• 2016
We prove a sequence of limiting results about weakly dependent stationary and regularly varying stochastic processes in discrete time. After deducing the limiting distribution for individual clustersExpand
Cluster based inference for extremes of time series
• Mathematics
• 2021
We introduce a new type of estimator for the spectral tail process of a regularly varying time series. The approach is based on a characterizing invariance property of the spectral tail process,Expand
On spectral gap properties and extreme value theory for multivariate affine stochastic recursions
• Mathematics
• 2016
We consider a general multivariate affine stochastic recursion and the associated Markov chain on \$\mathbb R^{d}\$. We assume a natural geometric condition which implies existence of an unboundedExpand

#### References

SHOWING 1-10 OF 60 REFERENCES
Convergence to Lévy stable processes under some weak dependence conditions
Abstract For a strictly stationary sequence of random vectors in R d we study convergence of partial sum processes to a Levy stable process in the Skorohod space with J 1 -topology. We identifyExpand
A functional limit theorem for dependent sequences with infinite variance stable limits
• Mathematics
• 2010
Under an appropriate regular variation condition, the affinely normalized partial sums of a sequence of independent and identically distributed random variables converges weakly to a non-GaussianExpand
Stable limits for sums of dependent infinite variance random variables
• Mathematics
• 2009
The aim of this paper is to provide conditions which ensure that the affinely transformed partial sums of a strictly stationary process converge in distribution to an infinite variance stableExpand
Extremal behaviour of models with multivariate random recurrence representation
• Mathematics
• 2007
For the solution Y of a multivariate random recurrence model Yn=AnYn-1+[zeta]n in we investigate the extremal behaviour of the process , , for with z*=1. This extends results for positive matricesExpand
Limit Theorems of Probability Theory
• Mathematics
• 2000
I. Classical-Type Limit Theorems for Sums of Independent Random Variables.- II. The Accuracy of Gaussian Approximation in Banach Spaces.- III. Approximation of Distributions of Sums of WeaklyExpand
Point Process and Partial Sum Convergence for Weakly Dependent Random Variables with Infinite Variance
• Mathematics
• 1995
Let {ξ j } be a strictly stationary sequence of random variables with regularly varying tail probabilities. We consider, via point process methods, weak convergence of the partial sums, S n = ξ 1Expand
Extremal behavior of regularly varying stochastic processes
• Mathematics
• 2005
We study a formulation of regular variation for multivariate stochastic processes on the unit interval with sample paths that are almost surely right-continuous with left limits and we provideExpand
On Large Deviations of Sums of Independent Random Variables
• Mathematics
• 2007
Extensions of some limit theorems are proved for tail probabilities of sums of independent identically distributed random variables satisfying the one-sided or two-sided Cramér's condition. The largeExpand
Precise large deviations for dependent regularly varying sequences
• Mathematics
• 2012
We study a precise large deviation principle for a stationary regularly varying sequence of random variables. This principle extends the classical results of Nagaev (Theory Probab Appl 14:51–64,Expand
A characterization of multivariate regular variation
• Mathematics, Economics
• 2000
We establish the equivalence between the multivariate regular variation of a random vector and the univariate regular variation of all linear combinations of the components of such a vector.Expand