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A quantitative central limit theorem for Poisson horospheres in high dimensions

Zakhar Kabluchko, Daniel Rosen, Christoph Thäle

Abstract

Consider a stationary Poisson process of horospheres in a $d$-dimensional hyperbolic space. In the focus of this note is the total surface area these random horospheres induce in a sequence of balls of growing radius $R$. The main result is a quantitative, non-standard central limit theorem for these random variables as the radius $R$ of the balls and the space dimension $d$ tend to infinity simultaneously.

A quantitative central limit theorem for Poisson horospheres in high dimensions

Abstract

Consider a stationary Poisson process of horospheres in a -dimensional hyperbolic space. In the focus of this note is the total surface area these random horospheres induce in a sequence of balls of growing radius . The main result is a quantitative, non-standard central limit theorem for these random variables as the radius of the balls and the space dimension tend to infinity simultaneously.
Paper Structure (4 sections, 4 theorems, 43 equations, 1 figure)

This paper contains 4 sections, 4 theorems, 43 equations, 1 figure.

Table of Contents

  1. Introduction and main result
  2. Proof of the main results
  3. Bounding $J_{{R,d}}$
  4. The Euclidean case

Key Result

Theorem 1

Let $N_{\hbox{$\frac{1}{2}$}}$ be a centred Gaussian random variable of variance $\frac{1}{2}$. Consider the surface functional $S_{R,d}$, for $d\geq 2$ and $R\geq 1$. Then there exists a universal constant $C > 0$ such that for any choice $\bullet \in \{\mathop{\mathrm{Kol}}\nolimits,\mathop{\mathr

Figures (1)

  • Figure 1: Simulation of a Poisson process of horospheres in the Poincaré disc model for the hyperbolic plane.

Theorems & Definitions (11)

  • Theorem 1
  • Remark 2
  • Lemma 3
  • proof
  • Remark 4
  • Proposition 5
  • proof
  • Lemma 6
  • Remark 7
  • proof : Proof of Theorem \ref{['thm:CLT-master']}
  • ...and 1 more