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Abstract
Poisson processes in the space of (d-1)-dimensional totally geodesic subspaces (hyperplanes) in a d-dimensional hyperbolic space of constant curvature -1 are studied. The k-dimensional Hausdorff measure of their k-skeleton is considered. Explicit formulas for first- and second-order quantities restricted to bounded observation windows are obtained. The central limit problem for the k-dimensional Hausdorff measure of the k-skeleton is approached in two different set-ups: (i) for a fixed window and growing intensities, and (ii) for fixed intensity and growing spherical windows. While in case (i) the central limit theorem is valid for all dge 2, it is shown that in case (ii) the central limit theorem holds for din {2,3} and fails if dge 4 and k=d-1 or if dge 7 and for general k. Also rates of convergence are studied and multivariate central limit theorems are obtained. Moreover, the situation in which the intensity and the spherical window are growing simultaneously is discussed. In the background are the Malliavin–Stein method for normal approximation and the combinatorial moment structure of Poisson U-statistics as well as tools from hyperbolic integral geometry.
Highlights
Background material and preparations3.1 More hyperbolic geometryRecall that by Hd we denote the hyperbolic space of dimension d
We explicitly identify the expectation and the covariance structure of these functionals by making recourse to general formulas for and structural properties of Poisson U-statistics and to Croftontype formulas from hyperbolic integral geometry
In addition and more importantly, we study probabilistic limit theorems for these functionals in the two asymptotic regimes described above for the Euclidean set-up
Summary
We denote by Hd , for d ≥ 2, the d-dimensional hyperbolic space of constant curvature −1, which is endowed with the hyperbolic metric dh( · , · ). We write ωk = 2π k/2/Γ (k/2), k ∈ N, for the surface area of the k-dimensional unit ball in the Euclidean space Rk. For t > 0, let ηt be a Poisson process on the space Ah(d, d − 1) of hyperplanes in Hd with intensity measure tμd−1. We refer to ηt as a (hyperbolic) Poisson hyperplane process with intensity t. Remark 1 In comparison with the Euclidean and spherical case we observe that precisely the same formula holds in these spaces This is not surprising, since the proof of Theorem 1 is based only on the multivariate Mecke formula for Poisson processes and a recursive application of Crofton’s formula from integral geometry, see Sect.
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