Extremal Theory for Stochastic Processes

Ledbetter, M R; Rootzén, Holger

doi:10.1214/aop/1176991767

Cited by 238 publications

(109 citation statements)

References 66 publications

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“…Relation (2.10) is close to the anti-clustering condition D (x a n ) used in extreme value theory; see Leadbetter et al [40], Leadbetter and Rootzén [39] and Embrechts et al [24,Chapter 5]. An alternative anti-clustering condition is (38) in Jakubowski [33]:…”

Section: Anti-clustering Conditionsmentioning

confidence: 67%

Stable limits for sums of dependent infinite variance random variables

Bartkiewicz

Jakubowski

Mikosch

et al. 2010

Probab. Theory Relat. Fields

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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 stable distribution. Conditions for this convergence to hold are known in the literature. However, most of these results are qualitative in the sense that the parameters of the limit distribution are expressed in terms of some limiting point process. In this paper we will be able to determine the parameters of the limiting stable distribution in terms of some tail characteristics of the underlying stationary sequence. We will apply our results to some standard time series models, including the GARCH(1, 1) process and its squares, the stochastic volatility models and solutions to stochastic recurrence equations.

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Section: Anti-clustering Conditionsmentioning

confidence: 67%

Stable limits for sums of dependent infinite variance random variables

Bartkiewicz

Jakubowski

Mikosch

et al. 2010

Probab. Theory Relat. Fields

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“…Here is a survey of the literature on the number of crossings of a given level or of a differentiable curve in a fixed time interval by a continuous spectrum Gaussian process. Besides the well-known books about Extremes, let us quote a short survey by Slud in 1994 (see [148] or [149]), a more general survey about extremes including a short section about level crossings by (see [86]) and by Rootzen in 1995 (see [132]), and another one by Piterbarg in his 1996 book (see [122], in particular for some methods described in more detail than here). Our purpose here is to focus only on crossing counts in order to be more explicit about the subject, not only recalling the main results, but also giving the main ideas about the different methods used to establish them.…”

Section: Crossings Of Gaussian Processesmentioning

confidence: 99%

Level crossings and other level functionals of stationary Gaussian processes

Kratz¹

2006

Probab. Surveys

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This paper presents a synthesis on the mathematical work done on level crossings of stationary Gaussian processes, with some extensions. The main results [(factorial) moments, representation into the Wiener Chaos, asymptotic results, rate of convergence, local time and number of crossings] are described, as well as the different approaches [normal comparison method, Rice method, Stein-Chen method, a general m-dependent method] used to obtain them; these methods are also very useful in the general context of Gaussian fields. Finally some extensions [time occupation functionals, number of maxima in an interval, process indexed by a bidimensional set] are proposed, illustrating the generality of the methods. A large inventory of papers and books on the subject ends the survey.

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“…Using the extremal theory of dependent processes, (see e.g. Leadbetter and Rootzen [20]), asymptotic properties of the (individual) extremes of (a 1 (x), . .…”

Section: µ(A ∩ B) − µ(A)µ(b)| ≤ Ce −λN µ(A)µ(b)mentioning

confidence: 99%