In a recent presentation, Christian Y. Robert provided a compact tour of Extreme Value Theory (EVT), emphasizing its importance for modeling heavy tails and rare events in fields such as finance, insurance, climate risk, engineering, and reliability.
Robert explained that standard statistical methods often fail for extreme events because these events are rare, poorly observed, and can exhibit strong dependencies across time, space, or variables. Instead, specialized probabilistic tools are required.
The talk covered univariate EVT fundamentals—maxima, record values, upper order statistics, and threshold exceedances—centered on the Generalized Extreme Value (GEV) and Generalized Pareto (GPD) distributions. The second part addressed multivariate extremes, including componentwise maxima, tail dependence, asymptotic dependence and independence, max-stable distributions, spectral measures, and stable tail dependence functions.
Robert then extended the discussion to infinite-dimensional settings, modeling extreme events as functions or spatiotemporal fields. This covered max-stable processes, spectral representations, generalized Pareto processes, and the radius-shape decomposition of extreme episodes.
Finally, he presented the point process viewpoint as a unifying framework for exceedances, Poisson limits, clustering of extremes, and extremal indices, linking EVT directly to generative modeling. A key takeaway: generative models for rare events must respect tail behavior, extremal dependence, and clustering, rather than merely reproducing typical observations.