Le mercredi 9 octobre 2024, l’Institut Fayol a eu le plaisir d’accueillir le Professeur Enkelejd Hashorva de l’Université de Lausanne, spécialiste renommé en théorie des valeurs extrêmes et des processus Gaussiens, dans le cadre du projet Ciroquo. C’est plus d’une trentaine de participants, en ligne et en présentiel, qui ont pu assister à son intervention.
Son intervention portait sur : « Extremes of Gaussian RF’s, Asymptotic Constants, Spectral Tail RF’s, Tail Measures & Cluster RF’s ».
Le Professeur Hashorva a exploré les propriétés des constantes asymptotiques, notamment les constantes de Pickands, ainsi que leur lien avec les champs aléatoires max-stables et les indices extrêmes des champs aléatoires stationnaires.
Abstract: This talk is motivated by the properties of certain asymptotic constants including the classical Pickands ones. Initially, Pickands constants appeared in the theory of extremes of stationary Gaussian processes. Later more general Pickands constants appeared in the analysis of the extremes of non-stationary random fields (rf’s) with light tails. Recent research has shown that generalised Pickands constants are closely related to stationary max-stable rf’s.
Moreover, there is a direct relation to extremal indices of stationary regularly varying rf’s. The latter are investigated via the theory of regular variation, which gave rise to the study of spectral tail and tail rf’s.
Alternatively, regular variation of rf’s has been also studied via tail measures, which are necessarily shift-invariant in the stationary case.
In this talk we shall discuss first recent results and challenges concerning extremes of Gaussian random field. For the framework of this talk, the stationarity of rf’s which gives rise to a particular shift-invariance property is crucial. We shall therefore define shift-invariant classes of jointly measurable and separable rf’s and show that those are the key to the definitions of spectral tail rf’s, cluster rf’s and shift-invariant tail measures.
Crucial for simulation or the estimation of various asymptotic constants of interest specific representations are required. It turns out that those representations have a specific meaning for related quantities such as shift-invariant tail measures or stationary max-stable rf’s.
We shall show that cluster rf’s are crucial for deriving various shift-representations of interest which lead for instance to knew Rosinski type representations for max-stable rf’s. Some new representations for generalised Pickands constants will also be elaborated.