A directional Poisson point process for univariate extremes

We state and prove the convergence of directionally-renormalised univariate random point measures to a limiting directional Poisson point process for extremes. The limiting point process forms an overarching characterisation of (asymptotically) small and large extremes drawn independently from a univariate probability measure and yields well-known joint limit laws for the minima and maxima of random samples as well as low and high threshold exceedances. • We characterise univariate extremes via a limiting directional Poisson point process. • The Poisson process is coherent with known limiting distributions for extremes. • We link the normalisation of the random point measures with the distribution of directions of the limit process.