Regularly varying multivariate time series

@article{Basrak2007RegularlyVM,
  title={Regularly varying multivariate time series},
  author={Bojan Basrak and J. Segers},
  journal={Stochastic Processes and their Applications},
  year={2007},
  volume={119},
  pages={1055-1080}
}
Extreme values of a stationary, multivariate time series may exhibit dependence across coordinates and over time. The aim of this paper is to offer a new and potentially useful tool called tail process to describe and model such extremes. The key property is the following fact: existence of the tail process is equivalent to multivariate regular variation of finite cuts of the original process. Certain remarkable properties of the tail process are exploited to shed new light on known results on… Expand

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References

SHOWING 1-10 OF 46 REFERENCES
Regular variation of GARCH processes
We show that the finite-dimensional distributions of a GARCH process are regularly varying, i.e., the tails of these distributions are Pareto-like and hence heavy-tailed. Regular variation of theExpand
Extremes of Markov chains with tail switching potential
Summary. A recent advance in the utility of extreme value techniques has been the characteri‐ zation of the extremal behaviour of Markov chains. This has enabled the application of extreme valueExpand
External Theory for Stochastic Processes.
Abstract : The purpose of this paper is to provide an overview of the asymptotic distributional theory of extreme values for a wide class of dependent stochastic sequences and continuous parameterExpand
The behavior of multivariate maxima of moving maxima processes
In the characterization of multivariate extremal indices of multivariate stationary processes, multivariate maxima of moving maxima processes, or M4 processes for short, have been introduced by SmithExpand
The sample autocorrelations of heavy-tailed processes with applications to ARCH
We study the sample ACVF and ACF of a general stationary sequence under a weak mixing condition and in the case that the marginal distributions are regularly varying. This includes linear andExpand
Extremal behaviour of models with multivariate random recurrence representation
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
Extremal Behaviour of Stationary Markov Chains with Applications
In this paper the extremal behaviour of real-valued, stationary Markov chains is studied under fairly general assumptions. Conditions are obtained for convergence in distribution of multilevelExpand
Multivariate regular variation of heavy-tailed Markov Chains
The upper extremes of a Markov chain with regulary varying stationary marginal distribution are known to exhibit under general conditions a multiplicative random walk structure called the tail chain.Expand
Moving averages with random coefficients and random coefficient autoregressive models
Consider the series ∑n C n Z n where {Z n} are iid -valued random vectors and {C n} are random matrices independent of the {Z n}. Under suitable summability conditions on the {C n}, if theExpand
Extremal behaviour of solutions to a stochastic difference equation with applications to arch processes
We consider limit distributions of extremes of a process {Yn} satisfying the stochastic difference equation Yn-AnYn-1+Bn, n[greater-or-equal, slanted]1,Y0[greater-or-equal, slanted]0, where {An, Bn}Expand
...
1
2
3
4
5
...