Existence and phase structure of random inverse limit measures

TL;DR

This paper establishes conditions for the existence and uniqueness of random inverse limit measures, exploring their phase structures and applying results to Dirichlet, Polya, and Gaussian families of random measures.

Contribution

It introduces a framework for understanding the existence, uniqueness, and phase structure of random inverse limit measures, extending classical theorems to measure-valued processes.

Findings

Four distinct phases of limiting random measures identified

Conditions for existence and uniqueness depend on topology and randomness notions

Applications to Dirichlet, Polya, and Gaussian measure families

Abstract

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). Results are applied in the well-known Dirichlet and Polya tree families of random probability measures and in 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 on defining parameters.