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High-dimensional limits arising from hyperbolic Poisson k-plane processes

Tillmann Bühler, Daniel Hug, Christoph Thäle

TL;DR

The paper analyzes high-dimensional limits for hyperbolic Poisson $k$-plane processes by examining the total $k$-volume of intersections within a ball and its infinitely divisible limit $Z_{d,k}$. It develops CLT-type criteria via the Lévy measure, aided by beta-function and incomplete beta-function estimates, to delineate when the variance-normalized limit is Gaussian or degenerate as $d\to\infty$ and $k$ grows with $d$. Moreover, it introduces a Lévy-measure rescaling that yields a non-Gaussian limit $\widetilde{Z}^{(b)}$ for fixed codimension $b=d-k$ and a Gaussian limit when $d-k\to\infty$, providing explicit density and cumulant formulas for the limiting non-Gaussian law. These results clarify the subtle high-dimensional behavior of geometric functionals in hyperbolic space, revealing Gaussian limits under certain scalings despite non-Gaussian finite-$d$ limits. The work thus bridges hyperbolic stochastic geometry with the theory of infinitely divisible distributions and offers explicit characterizations of limiting laws via Lévy densities.

Abstract

We consider a stationary Poisson process of $k$-planes in the $d$-dimensional hyperbolic space $\mathbb H^d$ of constant curvature $-1$, with $d \ge 4$ and $1 \le k \le d-1$. It is known that, after centring and normalization, the total $k$-volume of all intersections of $k$-planes with a geodesic ball of radius $R$ converges in distribution, as $R \to \infty$, to a non-Gaussian infinitely divisible random variable $Z_{d,k}$ whenever $2k > d+1$. We investigate the distributional behaviour of $Z_{d,k}$ in the high-dimensional regime $d \to \infty$ and depending on how fast $k$ grows in relation to $d$. We derive precise conditions for the variance normalized sequence to converge in law to a standard Gaussian random variable or to a degenerate law, respectively, and show that an alternative rescaling of the Lévy measures yields an explicit non-Gaussian infinitely divisible limit for fixed codimension $d-k$ and a standard Gaussian limit for $d-k \to \infty$.

High-dimensional limits arising from hyperbolic Poisson k-plane processes

TL;DR

The paper analyzes high-dimensional limits for hyperbolic Poisson -plane processes by examining the total -volume of intersections within a ball and its infinitely divisible limit . It develops CLT-type criteria via the Lévy measure, aided by beta-function and incomplete beta-function estimates, to delineate when the variance-normalized limit is Gaussian or degenerate as and grows with . Moreover, it introduces a Lévy-measure rescaling that yields a non-Gaussian limit for fixed codimension and a Gaussian limit when , providing explicit density and cumulant formulas for the limiting non-Gaussian law. These results clarify the subtle high-dimensional behavior of geometric functionals in hyperbolic space, revealing Gaussian limits under certain scalings despite non-Gaussian finite- limits. The work thus bridges hyperbolic stochastic geometry with the theory of infinitely divisible distributions and offers explicit characterizations of limiting laws via Lévy densities.

Abstract

We consider a stationary Poisson process of -planes in the -dimensional hyperbolic space of constant curvature , with and . It is known that, after centring and normalization, the total -volume of all intersections of -planes with a geodesic ball of radius converges in distribution, as , to a non-Gaussian infinitely divisible random variable whenever . We investigate the distributional behaviour of in the high-dimensional regime and depending on how fast grows in relation to . We derive precise conditions for the variance normalized sequence to converge in law to a standard Gaussian random variable or to a degenerate law, respectively, and show that an alternative rescaling of the Lévy measures yields an explicit non-Gaussian infinitely divisible limit for fixed codimension and a standard Gaussian limit for .

Paper Structure

This paper contains 7 sections, 10 theorems, 58 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background on infinitely divisible laws
  3. Estimates for beta functions
  4. Incomplete Beta function and Beta distribution
  5. Proofs
  6. Proof of Theorem \ref{['thm:CLT']}
  7. Proof of Theorem \ref{['thm:rescaling_levy_measure']}

Key Result

Theorem 1

Let $(d_n,k_n)_{n\geq 1}$ be a sequence satisfying $k_n < d_n$, $2k_n > d_n+1$ and $d_n \to \infty$ as $n\to\infty$. Write $Z^*_n \coloneqq Z_{d_n,k_n}/\sigma_{d_n,k_n}$ for the variance normalized version of $Z_{d_n,k_n}$ and abbreviate $r_n \coloneqq 2k_n - d_n - 1$.

Figures (1)

  • Figure 1: Densities of the random variables $\widetilde{Z}^{(b)}$ (top line) and densities of the Lévy measure $\nu^{(b)}_*$ (bottom line) with $b=1$ (left), $b=2$ (middle) and $b=3$ (right). The probability densities are obtained in a manner similar to that described in Section 6 of BH25.

Theorems & Definitions (17)

  • Theorem 1
  • Remark 2
  • Theorem 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 7 more