Stationary Min-Stable Stochastic Processes

An extreme-value theory model for dependent observations

“…Dr Isham touches on the issue of spatial extremes, from both a classical and a threshold point of view. In fact some models qualitatively similar to the Rodriguez-Iturbe, Cox and Isham models have also featured in theoretical representations for max-stable processes (de Haan and Pickands, 1986), and this motivates an approach somewhat similar to that suggested by Dr Isham. As with other extreme value procedures, an important feature is to concentrate on the extreme values rather than to try to fit a single model to the whole data set.…”

Minima of Independent Bessel Processes and of Distances Between Brownian Particles

“…If Y lt Y 2 ,..., Y n are independent copies of the process Y, then as we shall show below, the process (n 1/d min lij^n ^(n~2 /d /), t $s 0) has the same law as (Y(t),t ^ 0). This property of Y is similar to 'min-stability' as defined by de Haan and Pickands [6]; the difference is that here we must rescale time (to n~2 ld t) as well as space, to recover the original process.…”

Ergodic decompositions of stationary max-stable processes in terms of their spectral functions

“…By the measurability property, one may assume that the mapping (x, s) → ω x (s) is jointly measurable on X × S. According to de Haan and Pickands [5], see also [7] and [26], any stochastically continuous stationary max-stable process η admits a (distributional) representation of the form…”