Spectral tail processes and max-stable approximations of multivariate regularly varying time series

A regularly varying time series as introduced in Basrak and Segers (2009) is a (multivariate) time series such that all finite dimensional distributions are multivariate regularly varying. The extremal behavior of such a process can then be described by the index of regular variation and the so-called spectral tail process, which is the limiting distribution of the rescaled process, given an extreme event at time 0. As shown in Basrak and Segers (2009), the stationarity of the underlying time series implies a certain structure of the spectral tail process, informally known as the "time change formula". In this article, we show that on the other hand, every process which satisfies this property is in fact the spectral tail process of an underlying stationary max-stable process. The spectral tail process and the corresponding max-stable process then provide two complementary views on the extremal behavior of a multivariate regularly varying stationary time series

kk-means clustering of extremes

The $k$-means clustering algorithm and its variant, the spherical $k$-means clustering, are among the most important and popular methods in unsupervised learning and pattern detection. In this paper, we explore how the spherical $k$-means algorithm can be applied in the analysis of only the extremal observations from a data set. By making use of multivariate extreme value analysis we show how it can be adopted to find "prototypes" of extremal dependence and we derive a consistency result for our suggested estimator. In the special case of max-linear models we show furthermore that our procedure provides an alternative way of statistical inference for this class of models. Finally, we provide data examples which show that our method is able to find relevant patterns in extremal observations and allows us to classify extremal events

Invariance properties of limiting point processes and applications to clusters of extremes

Motivated by examples from extreme value theory we introduce the general notion of a cluster process as a limiting point process of returns of a certain event in a time series. We explore general invariance properties of cluster processes which are implied by stationarity of the underlying time series under minimal assumptions. Of particular interest are the cluster size distributions, where we introduce the two notions of inspected and typical cluster sizes and derive general properties of and connections between them. While the extremal index commonly used in extreme value theory is often interpreted as the inverse of a "mean cluster size", we point out that this only holds true for the expected value of the typical cluster size, caused by an effect very similar to the inspection paradox in renewal theory

Markov tail chains

The extremes of a univariate Markov chain with regulary varying stationary marginal distribution and asymptotically linear behavior are known to exhibit a multiplicative random walk structure called the tail chain. In this paper, we extend this fact to Markov chains with multivariate regularly varying marginal distribution in R^d. We analyze both the forward and the backward tail process and show that they mutually determine each other through a kind of adjoint relation. In a broader setting, it will be seen that even for non-Markovian underlying processes a Markovian forward tail chain always implies that the backward tail chain is Markovian as well. We analyze the resulting class of limiting processes in detail. Applications of the theory yield the asymptotic distribution of both the past and the future of univariate and multivariate stochastic difference equations conditioned on an extreme event

On Some Connections between Light Tails, Regular Variation and Extremes

In dieser Arbeit werden verschiedene Aspekte des Extremwertverhaltens von Verteilungen mit leichten und mit schweren Teils untersucht. Die Arbeit gliedert sich in drei Teile. Im ersten Teil werden Zusammenhänge zwischen Grenzwertsätzen für Summen und für Maxima von u.i.v. verteilten Zufallsvariablen hergestellt, indem Grenzwertsätze für $l_p$-Normen von positiven u.i.v. Zufallsvektoren untersucht werden. Ein neuer Ansatz ermöglicht es dabei, die Analyse für die unterschiedlichen Max-Anziehungsbereiche zu vereinheitlichen, wobei besonderes Interesse dem Gumbel-Fall gilt. Der zweite Teil der Arbeit beschäftigt sich mit dem Extremwertverhalten einer bestimmten Form von Zeitreihen, die eine asymptotische Ähnlichkeit zu sogenannten ``Random Difference Equations'' (RDEs) aufweisen. Wir erweitern ein Resultat, welches eine einfache Darstellung des Prozessverhaltens bedingt auf ein extremes Ereignis zum Zeitpunkt Null erlaubt, für eine einzelne Zeitreihe auf zwe! i zusammenhängende Zeitreihen und zeigen Anwendungsmöglichkeiten. Im dritten Teil der Arbeit wird das Extremalverhalten von RDEs in Bezug auf die Charakteristik $\kappa$, dem Index der regulären Variation, untersucht. Es wird eine neue Methode für die Bestimmung dieser Größe vorgeschlagen, die auf den Ergebnissen von Kesten (1973) beruht