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A central limit theorem for random tangent fields on stratified spaces

Jonathan C. Mattingly, Ezra Miller, Do Tran

Abstract

Variation of empirical Fréchet means on a metric space with curvature bounded above is encoded via random fields indexed by unit tangent vectors. A central limit theorem shows these random tangent fields converge to a Gaussian such field and lays the foundation for more traditionally formulated central limit theorems in subsequent work.

A central limit theorem for random tangent fields on stratified spaces

Abstract

Variation of empirical Fréchet means on a metric space with curvature bounded above is encoded via random fields indexed by unit tangent vectors. A central limit theorem shows these random tangent fields converge to a Gaussian such field and lays the foundation for more traditionally formulated central limit theorems in subsequent work.
Paper Structure (10 sections, 17 theorems, 59 equations)

This paper contains 10 sections, 17 theorems, 59 equations.

Table of Contents

  1. Prerequisites
  2. Random tangent fields
  3. Regularity of random tangent fields
  4. A Kolmogorov regularity theorem
  5. Hölder-continuous Gaussian tangent fields
  6. Chaining and covering: proof of Theorem \ref{['t:kolmogorov-reg']}
  7. CLT for random tangent fields
  8. Proof of the CLT for random tangent fields
  9. A priori and martingale bounds on Gn
  10. Convergence

Key Result

Lemma 1.2

The space of directions is a length space whose angular metric$\mathbf{d}_s$ satisfies

Theorems & Definitions (53)

  • Definition 1.1
  • Lemma 1.2
  • proof
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6: tangential-collapse
  • Definition 1.7
  • Remark 1.8
  • Definition 1.9: Stratified $\mathop{\mathrm{CAT}}\nolimits(\kappa)$ space
  • ...and 43 more