Extreme value theory for continuous parameter stationary processes View Full Text


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Article Info

DATE

1982-05

AUTHORS

M. R. Leadbetter, Holger Rootzén

ABSTRACT

In this paper the central distributional results of classical extreme value theory are obtained, under appropriate dependence restrictions, for maxima of continuous parameter stochastic processes. In particular we prove the basic result (here called Gnedenko's Theorem) concerning the existence of just three types of non-degenerate limiting distributions in such cases, and give necessary and sufficient conditions for each to apply. The development relies, in part, on the corresponding known theory for stationary sequences.The general theory given does not require finiteness of the number of upcrossings of any levelx. However when the number per unit time is a.s. finite and has a finite meanμ(x), it is found that the classical criteria for domains of attraction apply whenμ(x) is used in lieu of the tail of the marginal distribution function. The theory is specialized to this case and applied to give the general known results for stationary normal processes for whichμ(x) may or may not be finite).A general Poisson convergence theorem is given for high level upcrossings, together with its implications for the asymptotic distributions ofrth largest local maxima. More... »

PAGES

1-20

References to SciGraph publications

  • 1974-12. On extreme values in stationary sequences in PROBABILITY THEORY AND RELATED FIELDS
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    http://scigraph.springernature.com/pub.10.1007/bf01957094

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    http://dx.doi.org/10.1007/bf01957094

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