Lay summary

The classical Central Limit Theorem and its ramifications show that the Gaussian model is a natural and correct paradigm for building an approximate solution to many otherwise unsolvable problems encountered in various research fields.

Indeed, the range of applications of Gaussian processes and related random fields encompasses almost any field of theoretical and applied research. Some  extraordinary examples include variations of Brownian motion as the unique solution to problems from theoretical physics, biology, mathematical statistics, risk theory, stochastic finance, telecommunication,  just to name a few. While the theory of Gaussian processes and random fields is well-developed and mature, the range of applications of Gaussian random fields is constantly growing. Recently, applications in brain mapping, cosmology, and quantum chaos have been added to its palmares.  Due to the presence of measurement errors, missing observations or random inflations, in some cases the Gaussian framework appears as not tenable.

This project advocates that by extending the models to vector-valued chi-processes, vector-valued conditional Gaussian processes and random fields, the Gaussian framework proves to be very reliable. Essentially, numerous applications are intrinsically connected to the study of extremes of Gaussian processes and their related random fields. A natural extreme-value problem in this context is the determination of the exact tail asymptotic behaviour of the maxima of Gaussian processes over some given sets, the hardest and oldest problem in the study of random processes. Besides the tail asymptotics of the maximum, the derivation of limit theorems regarding the maxima of Gaussian processes is both of theoretical and applied interest.

This project aims at studying extremes of such large classes of vector-valued Gaussian processes, chi-processes, and conditional Gaussian processes over continuous, discrete and random sets. The principal theoretical findings envisaged by this study shall include both exact tail asymptotic results and limit theorems for the maxima of the mentioned Gaussian and related processes. Since real data are only possible to be observed on a certain discrete grid of time-points, it is planed to investigate the joint asymptotic behaviour of maximum over continuous time intervals with maxima over discrete grids, for several classes of Gaussian processes and chi-processes. Motivated by various applications in risk theory, queueing theory, and hydrodynamics this project is also concerned with
the study of the maximum of Gaussian processes and chi-processes over random time intervals. In addition to numerous theoretical results and their interpretation, this project shall develop novel methodologies and techniques. Furthermore, the derivation of some key asymptotical results for the extremes of several Gaussian fields will open the way for novel statistical applications, whereas by focusing on both Gaussian perturbed risk processes and generalsations of the storage processes, additional applications concerned with the risk analysis, simulation of rare-events and the analysis of overflows in hydrodynamics will be promoted.