Existence and Phase Structure of Random Inverse Limit Measures

Analogous to Kolmogorov’s theorem for the existence of stochastic processes describing random functions, we consider theorems for the existence of stochastic processes describing random measures as limits of inverse measure systems. Specifically, given a coherent inverse system of random (bounded/signed/positive/probability) histograms on refining partitions, we study conditions for the existence and uniqueness of a corresponding random inverse limit, a Radon probability measure on the space of (bounded/signed/positive/probability) measures. Depending on the topology (vague/tight/weak/total-variational) and Kingman’s notion of complete randomness, the limiting random measure is in one of four phases, distinguished by their degrees of concentration (support/domination/discreteness). The results are applied in the well-known Dirichlet and Polya tree families of random probability measures and a new Gaussian family of signed inverse limit measures. In these three families, examples of all four phases occur, and we describe the corresponding conditions of defining parameters.