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Uniform Continuity in Distribution for Borel Transformations of Random Fields

Alexander I. Bufetov

TL;DR

The paper addresses when Borel transformations of random-field realizations depend uniformly continuously on the driving law. It builds a general framework with a Polish space $W$, the probability measures ${\mathfrak{M}}_1(W)$, and a compact subfamily ${\mathfrak{N}} \subset {\mathfrak{M}}_1(W)$ to study pushforwards $g_*$. The main approach reduces to the case $W=\mathbb{C}^{\mathbb{N}}$ and establishes a uniform continuity result (Theorem 2.3) by analyzing finite-dimensional marginals and using compactness and projection arguments, including the Bockstein theorem to justify reduction to countable coordinates. A constructive sufficient condition for separability of the space of realizations is provided via $L^2$-type bounds on coordinate maps and a countable dense index set, yielding a robust setting for limit arguments; the paper also connects to Gaussian multiplicative chaos as an important potential application for future work. These results facilitate robust, coordinate-free analysis of distributions of transformed random fields and set the stage for further study of GMC-type functionals.

Abstract

Simple sufficient conditions are given that ensure the uniform continuity in distribution for Borel transformations of random fields.

Uniform Continuity in Distribution for Borel Transformations of Random Fields

TL;DR

The paper addresses when Borel transformations of random-field realizations depend uniformly continuously on the driving law. It builds a general framework with a Polish space , the probability measures , and a compact subfamily to study pushforwards . The main approach reduces to the case and establishes a uniform continuity result (Theorem 2.3) by analyzing finite-dimensional marginals and using compactness and projection arguments, including the Bockstein theorem to justify reduction to countable coordinates. A constructive sufficient condition for separability of the space of realizations is provided via -type bounds on coordinate maps and a countable dense index set, yielding a robust setting for limit arguments; the paper also connects to Gaussian multiplicative chaos as an important potential application for future work. These results facilitate robust, coordinate-free analysis of distributions of transformed random fields and set the stage for further study of GMC-type functionals.

Abstract

Simple sufficient conditions are given that ensure the uniform continuity in distribution for Borel transformations of random fields.
Paper Structure (4 sections, 12 theorems, 41 equations)

This paper contains 4 sections, 12 theorems, 41 equations.

Table of Contents

  1. Introduction and the statement of the main result
  2. Reduction to the case $W = \mathbb{C}^{\mathbb{N}}$
  3. Proof of Theorem \ref{['thm:2.3']}
  4. A sufficient condition for separability of the space of realizations

Key Result

Theorem 1.1

The correspondence $g \mapsto g_*$ induces a uniformly continuous map from the space $\mathscr{B}_{\widetilde{{\mathfrak{N}}}}({\mathfrak{N}} \times V, W)$ to the space $\mathscr{B}({\mathfrak{N}}, \widetilde{{\mathfrak{N}}})$.

Theorems & Definitions (15)

  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Proposition 2.5
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • Corollary 3.3
  • ...and 5 more