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Extremal behavior of regularly varying stochastic processes
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.ORCID iD: 0000-0001-9210-121X
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
2005 (English)In: Stochastic Processes and their Applications, ISSN 0304-4149, Vol. 115, no 2, 249-274 p.Article in journal (Refereed) Published
Abstract [en]

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 provide necessary and sufficient conditions for such stochastic processes to be regularly varying. A version of the Continuous Mapping Theorem is proved that enables the derivation of the tail behavior of rather general mappings of the regularly varying stochastic process. For a wide class of Markov processes with increments satisfying a condition of weak dependence in the tails we obtain simplified sufficient conditions for regular variation. For such processes we show that the possible regular variation limit measures concentrate on step functions with one step, from which we conclude that the extremal behavior of such processes is due to one big jump or an extreme starting point. By combining this result with the Continuous Mapping Theorem, we are able to give explicit results on the tail behavior of various vectors of functionals acting on such processes. Finally, using the Continuous Mapping Theorem we derive the tail behavior of filtered regularly varying Levy processes.

Place, publisher, year, edition, pages
2005. Vol. 115, no 2, 249-274 p.
Keyword [en]
regular variation, extreme values, functional limit theorem, Markov, prooesses, subadditive functionals
National Category
Probability Theory and Statistics
URN: urn:nbn:se:kth:diva-14477DOI: 10.1016/ 000226563000003OAI: diva2:332518
QC 20100525Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2012-03-07Bibliographically approved

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