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

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arxiv 2402.14331 v1 pith:XFDKOPTM submitted 2024-02-22 math.MG math.PR

Syota Esaki , Daisuke Kazukawa , Ayato Mitsuishi This is my paper · ORCID
classification math.MGmath.PR
keywords convergesspacesapplicationcauchyconcentrationconesdistributionmeasure
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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.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Pyramidal Compactification of Asymmetric Metric Measure Spaces via Adjoint Transport

    math.MG 2026-08 conditional novelty 7.0 of 10

    A compactification of pyramids of asymmetric metric measure spaces is constructed using one-sided observables and adjoint transport, preserving directed endpoints.

  2. Poincar\'e Beta Balls: Radial Laws, Shell Transforms, and Phase Diagram

    math.MG 2026-07 accept novelty 7.0 of 10

    Rescaled Poincaré beta balls converge weakly to one of four pyramids—finite star trees, diameter ≤1, metric-transformed Gaussians, or the Gaussian pyramid—according to the A and βL balance.

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