Graphical models for infinite measures with applications to extremes

Conditional independence and graphical models are well studied for probability distributions on product spaces. We propose a new notion of conditional independence for any measure $\Lambda$ on the punctured Euclidean space $\mathbb R^d\setminus \{0\}$ that explodes at the origin. The importance of such measures stems from their connection to infinitely divisible and max-infinitely divisible distributions, where they appear as L\'evy measures and exponent measures, respectively. We characterize independence and conditional independence for $\Lambda$ in various ways through kernels and factorization of a modified density, including a Hammersley-Clifford type theorem for undirected graphical models. As opposed to the classical conditional independence, our notion is intimately connected to the support of the measure $\Lambda$. Our general theory unifies and extends recent approaches to graphical modeling in the fields of extreme value analysis and L\'evy processes. Our results for the corresponding undirected and directed graphical models lay the foundation for new statistical methodology in these areas.