Séance Séminaire

Séminaire de Probabilités et Statistique

On some multivariate crossing problems.

We aim to investigate theoretical and statistical properties related to some level functionals in the context of stationary random processes and fields. A variety of phenomena in physical and biological sciences can be mathematically understood by considering the statistical properties of level-crossings of random processes or fields. Notably, a growing number of these phenomena require a consideration of correlated level-crossings emerging from multiple correlated processes. While many theoretical results have been obtained in the last decades for individual Gaussian level-crossing processes, few results are available for multivariate, jointly correlated threshold crossings. In this chapter we will focus essentially on this latter point. In particular, we present an application in neuroscience by studying up-crossing of multivariate, jointly correlated neuronal voltages. Furthermore, a Gaussianity test is proposed by using again some level functionals of a single realization of a random field and observations exceeding certain thresholds. We also focus on non-continuous random fields, like some types of shot noise processes and we study the perimeter of excursion sets and the mean excursion EC of bivariate shot noise random fields. A related statistical analysis based on these Minkowski functionals is the object of a working paper.