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Existence and phase structure of random inverse limit measures

B. J. K. Kleijn

TL;DR

The paper develops a comprehensive existence theory for random measures as inverse limits of random histogram systems on refining partitions, extending Kolmogorov-type constructions to spaces of measures $M^1({\mathscr X})$ under tight, weak, and total-variation topologies. Using coherence and the Bourbaki-Prokhorov-Schwartz framework, it provides necessary and sufficient conditions for the existence and uniqueness of Radon limits, and identifies a four-phase structure (absolutely-continuous, fixed-atomic, continuous-singular, random-atomic) that can manifest under different topologies and dependence structures. The theory is illustrated across three major families: Dirichlet histogram limits, Pólya-tree histogram limits, and Gaussian histogram limits (including signed measures), with detailed results on when tight or weak limits exist and which phase they inhabit. The work emphasizes the constructive and computational aspects of histogram limits, enabling finite-dimensional simulations and offering potential applications in nonparametric Bayes, stochastic analysis, and connections to random field theory. Overall, it unifies and extends existing random-measure constructions, providing new tools to analyze phase transitions and approximations of random measures.

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.

Existence and phase structure of random inverse limit measures

TL;DR

The paper develops a comprehensive existence theory for random measures as inverse limits of random histogram systems on refining partitions, extending Kolmogorov-type constructions to spaces of measures under tight, weak, and total-variation topologies. Using coherence and the Bourbaki-Prokhorov-Schwartz framework, it provides necessary and sufficient conditions for the existence and uniqueness of Radon limits, and identifies a four-phase structure (absolutely-continuous, fixed-atomic, continuous-singular, random-atomic) that can manifest under different topologies and dependence structures. The theory is illustrated across three major families: Dirichlet histogram limits, Pólya-tree histogram limits, and Gaussian histogram limits (including signed measures), with detailed results on when tight or weak limits exist and which phase they inhabit. The work emphasizes the constructive and computational aspects of histogram limits, enabling finite-dimensional simulations and offering potential applications in nonparametric Bayes, stochastic analysis, and connections to random field theory. Overall, it unifies and extends existing random-measure constructions, providing new tools to analyze phase transitions and approximations of random measures.

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.
Paper Structure (34 sections, 34 theorems, 140 equations, 2 figures)

This paper contains 34 sections, 34 theorems, 140 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Limits of random histogram systems
  3. Inverse systems of random histograms
  4. Measures, partitions and histograms
  5. Domination, histogram densities and total variation
  6. Random histogram systems and coherence
  7. The Bourbaki-Prokhorov-Schwartz theorem
  8. Random histogram limits with the weak topology
  9. Support and approximation of weak histogram limits
  10. Support and domination
  11. Approximation by weakly convergent histograms
  12. Existence of weak histogram limits
  13. Existence of total-variational histogram limits
  14. Random histogram limits with the tight topology
  15. Existence of tight histogram limits
  16. ...and 19 more sections

Key Result

Lemma 2.5

Let ${\mathscr X}$, ${\mathscr A}$ and $M^1({\mathscr X})$ satisfy the minimal conditions and assume that ${\mathscr A}$ resolves ${\mathscr X}$. Then, for any $P\in M^1({\mathscr X})$ and any dominating $Q\in M^1({\mathscr X})$, $P_{Q,{\alpha}}$ converges to $P$ in total variation.

Figures (2)

  • Figure 1: A sample from a random histogram on a 64x64 partitioned square patch of two-dimensional Euclidean space-time with the Green's function for the Laplacian to define the covariance measure, and its 32x32, 16x16 and 8x8 coarsened histograms. Coherence of the histogram system says that the distributions of the random 8x8, 16x16 and 32x32 histograms must equal the distributions implied by coarsening of the random 64x64 histogram. The histogram limit is the random object obtained by infinite refinement, to the infinite right of these four histograms.
  • Figure 2: Samples from Gaussian random histograms on a 64x64 partitioned square slice of Euclidean space-time, with the Green's function for the Laplacian to define the covariance measure, in two dimensions; in three dimensions; and in four dimensions; alongside a sample with the Yukawa potential of a massive scalar boson field in four dimensions.

Theorems & Definitions (62)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Lemma 2.5
  • Proposition 2.6
  • Lemma 2.7
  • Proposition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 52 more