In a recent presentation at the Generative AI for Extreme Events conference, Part III, Christian Y. Robert delivered a comprehensive overview of Extreme Value Theory (EVT) and its critical role in modeling rare, high-impact events. The talk emphasized that standard statistical methods often fail when dealing with heavy tails and rare occurrences, making specialized probabilistic tools essential.
Robert began by motivating the discussion with real-world applications in finance, insurance, climate risk, engineering, and reliability—fields where extreme events are infrequent, poorly observed, and often exhibit strong dependencies across time, space, or variables. He then outlined the foundational concepts of univariate EVT, including maxima, record values, upper order statistics, and threshold exceedances, highlighting the central role of the Generalized Extreme Value (GEV) and Generalized Pareto (GPD) distributions.
The second part shifted to multivariate extremes, covering componentwise maxima, tail dependence, asymptotic dependence and independence, max-stable distributions, spectral measures, stable tail dependence functions, and multivariate generalized Pareto distributions. These tools are crucial for understanding how extreme events correlate across multiple variables.
Extending to infinite-dimensional settings, the third part addressed functional and spatial-temporal extremes. Robert explained max-stable processes, spectral representations, generalized Pareto processes, and the radius-shape decomposition of extreme episodes. These frameworks enable modeling of extreme events as continuous functions or fields.
Finally, the presentation introduced the point process viewpoint, which unifies exceedances, Poisson limits, clustering of extremes, extremal indices, and connections to generative modeling. A key takeaway was that generative models for rare events must not merely reproduce training data but should extrapolate to unseen, yet plausible, extreme scenarios—a challenge that EVT is uniquely equipped to address.