Intersections of Poisson k-flats in hyperbolic space: Completing the picture

Tillmann Bühler and Daniel Hug

Stochastic Processes and their Applications, 2025, vol. 185, issue C

Abstract: Let η be an isometry invariant Poisson process of k-flats, 0≤k≤d−1, in d-dimensional hyperbolic space. For d−m(d−k)≥0, the m-th order intersection process of η consists of all nonempty intersections of distinct flats E1,…,Em∈η. Of particular interest is the total volume Fr(m) of this intersection process in a ball of radius r. For 2k>d+1, we determine the asymptotic distribution of Fr(m), as r→∞, previously known only for m=1, and derive rates of convergence in the Kolmogorov distance. Properties of the non-Gaussian limit distribution are discussed. We further study the asymptotic covariance matrix of the vector (Fr(1),…,Fr(m))⊤.

Keywords: Poisson process; Hyperbolic space; k-flat process; Limit theorem; Kolmogorov distance; Berry–Esseen bound (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1016/j.spa.2025.104613

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