On the continuity of Pickands constants

Abstract

For a non-negative separable random field Z(t), t ∈ R d satisfying some mild assumptions we show thatfor δ ≥ 0 where 0Z d := R d and prove that H 0 Z can be approximated by H δ Z if δ tends to 0. These results extend the classical findings for the Pickands constants H δ Z , defined for Z(t) = exp √ 2Bα(t) − |t| 2α , t ∈ R with Bα a standard fractional Brownian motion with Hurst parameter α ∈ (0, 1]. The continuity of H δ Z at δ = 0 is additionally shown for two particular extensions of Pickands constants.

Shift-invariant homogeneous classes of random fields

“…Let hereafter K α [Z] be a given shift-invariant α-homogeneous class of rf's and Θ, Y it spectral and tail rf's, respectively. The next claim is a consequence of (4.2) and is based on similar calculations as in [52]…”

Uniform bounds for ruin probability in Multidimensional Risk Model

“…We are interested in the calculation of the finite-time ruin probability for given T > 0. The inequalities below have already been shown in [2]. We retrieve them using our findings.…”

“…Inequality (1) has been crucial in the context of Shepp-statistics investigated in [1]. It is also of great importance in the investigation of simultaneous ruin probabilities in vector-valued risk models (see [2][3][4]).…”

“…It is of interest to consider a general trend function in (2). We consider below a large class of trend functions which is tractable if Z has self-similar coordinates with index α > 0.…”