Pickands constants play an important role in the exact asymptotic of extreme values for Gaussian stochastic processes.

No abstract

Let {X(t) : t ∈ [0, ∞)} be a centered Gaussian process with stationary increments and variance function σ 2 X (t). We study the exact asymptotics of P(sup t∈[0,T ] X(t) > u) as u → ∞, where T is an independent of {X(t)} non-negative Weibullian random variable. As an illustration, we work out the asymptotics of the supremum distribution of fractional Laplace motion. This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2011, Vol. 17, No. 1, 194-210. This reprint differs from the original in pagination and typographic detail.

This paper derives an exact asymptotic expression forwhere X(t) = (X 1 (t), . . . , X d (t)) ⊤ , t ≥ 0 is a correlated d-dimensional Brownian motion starting at the pointThe derived asymptotics depends on the solution of an underlying multidimensional quadratic optimization problem with constraints, which leads in some cases to dimension-reduction of the considered problem. Complementary, we study asymptotic distribution of the conditional first passage time to U, which depends on the dimension-reduction phenomena.