Statistical inference for radial generalized Pareto distributions and return sets in geometric extremes

We use a functional analogue of the quantile function for probability measures on $\mathbb{R}^d$ to characterize a novel limit Poisson point process for radially recentred and rescaled random vectors under a radial-directional decomposition. This limit process yields new multivariate distributions, including \textit{radial generalised Pareto distributions}, exhibiting stability for extrapolation to extremal sets along any direction. We show that the normalising functions leading to the limit Poisson point process correspond to a novel class of sets visited with fixed probability, with geometric properties determined by the conditional distribution of the radius given the direction and the Radon-Nikodym derivative of the directional probability distribution relative to reference spherical measures. This leads to return sets, defined by the complement of these probability sets and expressed by their return period. We identify an important member, the \textit{isotropic return set}, where all directions of exceedances outside the set are equally likely. Building on the limit Poisson point process likelihood, we develop parsimonious statistical models leveraging links between limit distribution parameters, with novel diagnostics for assessing convergence to the limiting distribution. These models enable Bayesian inference for return sets with arbitrarily large return periods and probabilities of unobserved extreme events, incorporating directional information from observations outside probability sets. The framework supports efficient computations in dimensions d=2 and d=3. We demonstrate the utility of the methods through simulations and case studies involving hydrological and oceanographic data, showcasing potential for robust and interpretable analysis of multivariate extremes.