Grant: Discovery Grants Program - Individual

Understanding and predicting extreme events, as well as their impact on society, is vital. Extremes appear in many applications such as finance, environmental sciences, random graphs and networks. The appropriate mathematical theory is known as Extreme Value Theory (EVT), a branch of probability and statistics. It traditionally deals with independent data. It is naturally linked to heavy-tailed analysis. Later developments rendered EVT capable of handling (low-dimensional) multivariate data, stationary time series (so-called extremal clustering) or spatial phenomena. However, with increased prevalence of high-dimensional and complex data, available methods from EVT became too limited. The goal of this research program is to develop the probabilistic theory and tools for statistical inference for extremes of complex data structures, with a focus on four subtopics. 1) Non-stationary time series. This line of research is motivated by environmental applications. The existing results focus on independent, non-identically distributed random variables. We will develop the probabilistic foundations for extremes of non-stationary time series. We will then link the probabilistic theory to statistical inference and will develop limit theorems for appropriately constructed blocks estimators of different indices that quantify time-varying extremal clustering and tail behaviour. 2) Functional time series. This line of research is motivated by financial and environmental applications. The existing theory is very limited. The main challenge stems from the fact that the existing tools for functional time series are based on a Hilbert space analysis which is not appropriate in the extreme context. As such, EVT for stochastic processes in Banach spaces will need to be developed. The probabilistic theory will be linked to statistical inference - we will estimate the indices that quantify extremal functional clustering. 3) Continuous processes. High frequency financial data can be also studied with help of continuous time processes. Recent developments in their probabilistic properties should allow to conduct the statistical inference, similar to discrete time-series. The main issue is the sampling frequency, that causes an additional bias. 4) Random graphs. This topic is motivated by social networks. They are very inhomogeneous in nature, with few people having many connections. This can be naturally modelled by heavy tails. The associated tail index is of a primary interest. While some theoretical work has been done for extremes of so-called preferential attachment networks, little work has been done for extremes of general, inhomogeneous random graphs. The main challenge stems from the fact that time-dependence structure is lost and completely new tools have to be developed. Our research extends the limits of EVT to empower end-users with better predictions and decisions regarding extreme events in diverse applications.