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Convergence of cones of metric measure spaces and its application to Cauchy distribution

Syota Esaki, Daisuke Kazukawa, Ayato Mitsuishi

Abstract

We prove that the sequence of cones of metric measure spaces converges if the sequence of base spaces converges in Gromov's box, concentration, and weak topologies. As an application, we show that the generalized Cauchy distribution with suitable scaling converges to a half line in the concentration topology as the dimension diverges to infinity. This is a new example distinguished from previously known examples such as Gaussian distributions and typical closed Riemannian manifolds with constant Ricci curvature.

Convergence of cones of metric measure spaces and its application to Cauchy distribution

Abstract

We prove that the sequence of cones of metric measure spaces converges if the sequence of base spaces converges in Gromov's box, concentration, and weak topologies. As an application, we show that the generalized Cauchy distribution with suitable scaling converges to a half line in the concentration topology as the dimension diverges to infinity. This is a new example distinguished from previously known examples such as Gaussian distributions and typical closed Riemannian manifolds with constant Ricci curvature.
Paper Structure (13 sections, 40 theorems, 190 equations)

This paper contains 13 sections, 40 theorems, 190 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Metric measure spaces
  4. Box distance and observable distance
  5. Lévy family
  6. Pyramid
  7. Dissipation
  8. Convergence of cones
  9. Another simple proof for Lévy families
  10. Cauchy distribution
  11. Proof of Theorem \ref{['thm:non-box']}
  12. Several quantities for Cauchy space
  13. Relation to the ergodic theory

Key Result

Theorem 1.2

Let $\kappa \in \mathbb{R}$. Assume that a sequence $\{\mathcal{P}_n\}_{n=1}^\infty$ of pyramids converges weakly to a pyramid $\mathcal{P}$ and that a sequence $\{\mu_n\}_{n=1}^\infty$ of Borel probability measures on $I_\kappa$ converges weakly to a Borel probability measure $\mu$. Then the sequen

Theorems & Definitions (91)

  • Definition 1.1: $\kappa$-Cone
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Definition 2.1: mm-Isomorphism
  • Definition 2.2: Lipschitz order
  • ...and 81 more