LMS Invited Lectures 2026: Extreme Value Theory for Heavy-Tailed Time Series

Accompanying Lectures

TBA - Prof. Jennifer Wadsworth (Lancaster University).

Abstract: TBA

Domain-scaled regular variation - Mathematical foundations for a new tail approximation - Dr. Kirstin Strokorb (University of Bath).

Abstract: The concept of regular variation has a long history in probability and statistics. In the context of regularly varying spatial stochastic processes part of the mathematical appeal stems from the equivalence of multiple characterizations of regular variation, which in turn can be exploited for statistical inference and tail extrapolation. On the other hand, the asymptotic stability properties of a regularly varying process are rather rigid and are appropriate only for the modelling of the tail behaviour of so-called asymptotically dependent stochastic processes. Instead, empirical evidence suggests that spatial extremes often exhibit weakening of dependence that is not compatible with pure asymptotic dependence, that is, they become more localized as the conditioning threshold increases. In this talk I will introduce a refined notion of regular variation that overcomes this limitation via domain-scaling. Our theory is inspired by the triangular array convergence of domain-scaled maxima of Gaussian processes to a Brown–Resnick process.

We shall see that this modification via domain-scaling can conveniently be incorporated into the theory of regular variation for spatial stochastic processes to make it more flexible and better reflect empirical evidence. We study key properties of the resulting tail approximating process and demonstrate its ability to approximate conditional exceedance probabilities of Gaussian processes. Mathematical convenience arises from the recently rediscovered concept of vague convergence based on boundedness. This talk is based on joint work with Marco Oesting and Raphaël de Fondeville (preprint https://doi.org/10.14760/OWP-2025-02).

Self-normalization of sums of dependent random variables - Prof. Olivier Wintenberger (LPSM, Sorbonne University).

Abstract: In this talk, we analyze the extremal dependence for regularly varying stationary time series. After introducing the extremogram as a specific tool to visualize the length of memory for extreme values, we study the joint limit behavior of sums, maxima, and standard deviations of samples. As a consequence, we can determine the distributional limits for ratios of sums and maxima, as well as studentized sums, even when the variance is infinite. We calculate moments of the limits and point out the differences between the iid (independent and identically distributed) and some other weakly dependent time series. We highlight some unknown side effects of the clustering of extremes in common ratio statistics.

Based on a collaboration with M. Matsui (Osaka) and T. Mikosch (Copenhagen)