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Contents of PMS, Vol. 37, Fasc. 2,
pages 299 - 317
 

ON EXTREMAL INDEX OF MAX-STABLE STATIONARY PROCESSES

Krzysztof Dębicki
Enkelejd Hashorva

Abstract: In this contribution we discuss the relation between Pickands-type constants defined for certain Brown–Resnick stationary process W (t),t ∈ ℝ , as

H δ = lim T-1E( sup eW (t)), δ ≥ 0 W T→ ∞ t∈δℤ∩[0,T]

(set 0ℤ = ℝ if δ = 0 ) and the extremal index of the associated max-stable stationary process ξW . We derive several new formulas and obtain lower bounds for H δW if W is a Gaussian or a Lévy process. As a by-product we show an interesting relation between Pickands constants and lower tail probabilities for fractional Brownian motions.

2010 AMS Mathematics Subject Classification: Primary: 60G15; Secondary: 60G70.

Keywords and phrases: Extremal index, mean cluster index, Pickands constant, M3 representation, Brown–Resnick stationary, max-stable process, Gaussian process, Lévy process.

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