Generalized Pickands constants and stationary max-stable processes

Abstract

Abstract. Pickands constants play a crucial role in the asymptotic theory of Gaussian processes. They are commonly defined as the limits of a sequence of expectations involving fractional Brownian motions and, as such, their exact value is often unknown. Recently, Dieker and Yakir (2014) derived a novel representation of Pickands constant as a simple expected value that does not involve a limit operation. In this paper we show that the notion of Pickands constants and their corresponding Dieker-Yakir representations can be extended to a large class of stochastic processes, including general Gaussian and Lévy processes. We furthermore provide a link to spatial extreme value theory and show that Pickands-type constants coincide with certain constants arising in the study of max-stable processes with mixed moving maxima representations.

Approximation of ruin probability and ruin time in discrete Brownian risk models

“…As in the proof of Theorem 1.1 we obtain and ω(j, S, x) is defined in (17). By Borell-TIS inequality (similarly the proof of (18) It follows with similar arguments as in [24] that as S → ∞…”

Approximation of Sojourn Times of Gaussian Processes

“…In Theorem 1.1 we present an interesting formula for H α , which is a consequence of Berman's theory on extremes of random processes. We believe that this new formula is of particular interest for simulations, since it is given as an expectation, see [7][8][9][10][11] for alternative formulas. Another advantage of this new formula is that it implies the uniformly (with respect to α) sharpest lower bound for the Pickands constant available in the literature so far.…”

Extremes of threshold-dependent Gaussian processes

“…The finiteness of P f α,a [0, ∞) and P f α,a (−∞, ∞) is guaranteed under weak assumptions on f , which will be shown in the proof of Theorem 2.2, see [2,3,5,7,15,25,[39][40][41][42][43] for various properties of H α and P f α,a [0, ∞). Denote by I {·} the indicator function.…”

“…are the Pickands and Piterbarg constants, respectively, where B α is a standard fractional Brownian motion (fBm) with self-similarity index α/2 ∈ (0, 1], see [19][20][21][22][23][24][25] for properties of both constants.…”