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Christoph Thäle

Publications and source records attributed to Christoph Thäle.

At least 37 records · Page 2Linked to original sources
arXiv2024

Intersection probabilities for flats in hyperbolic space

Consider the $d$-dimensional hyperbolic space $\mathbb{M}_K^d$ of constant curvature $K<0$ and fix a point $o$ playing the role of an origin. Let $\mathbf{L}$ be a uniform random $q$-dimensional totally geodesic submanifold (called $q$-flat) in $\mathbb{M}_K^d$ passing through $o$ and, independently of $\mathbf{L}$, let $\mathbf{E}$ be a random $(d-q+γ)$-flat in $\mathbb{M}_K^d$ which is uniformly distributed in the set of all $(d-q+γ)$-flats intersecting a hyperbolic ball of radius $u>0$ around $o$. We are interested in the distribution of the random $γ$-flat arising as the intersection of $\mathbf{E}$ with $\mathbf{L}$. In contrast to the Euclidean case, the intersection $\mathbf{E}\cap \mathbf{L}$ can be empty with strictly positive probability. We determine this probability and the full distribution of $\mathbf{E}\cap \mathbf{L}$. Thereby, we elucidate crucial differences to the Euclidean case. Moreover, we study the limiting behaviour as $d\uparrow\infty$ and also $K\uparrow 0$. Thereby we obtain a phase transition with three different phases which we completely characterize, including a critical phase with distinctive behavior and a phase recovering the Euclidean results. In the background are methods from hyperbolic integral geometry.

math.PR↗
arXiv2024

A quantitative central limit theorem for Poisson horospheres in high dimensions

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.

math.PR↗
arXiv2024

Fluctuations of $λ$-geodesic Poisson hyperplanes in hyperbolic space

Poisson processes of so-called $λ$-geodesic hyperplanes in $d$-dimensional hyperbolic space are studied for $0\leqλ\leq 1$. The case $λ=0$ corresponds to genuine geodesic hyperplanes, the case $λ=1$ to horospheres and $λ\in(0,1)$ to $λ$-equidistants. In the focus are the fluctuations of the centred and normalized total surface area of the union of all $λ$-geodesic hyperplanes in the Poisson process within a hyperbolic ball of radius $R$ centred at some fixed point, as $R\to\infty$. It is shown that for $λ<1$ these random variables satisfy a quantitative central limit theorem precisely for $d=2$ and $d=3$. The exact form of the non-Gaussian, infinitely divisible limiting distribution is determined for all higher space dimensions $d\geq 4$. The special case $λ=1$ is in sharp contrast to this behaviour. In fact, for the total surface area of Poisson processes of horospheres, a non-standard central limit theorem with limiting variance $1/2$ is established for all space dimensions $d\geq 2$. We discuss the analogy between the problem studied here and the Random Energy Model whose partition function exhibits a similar structure of possible limit laws.

math.PR↗
arXiv2024

Quantification of the Fourth Moment Theorem for Cyclotomic Generating Functions

This paper deals with sequences of random variables $X_n$ only taking values in $\{0,\ldots,n\}$. The probability generating functions of such random variables are polynomials of degree $n$. Under the assumption that the roots of these polynomials are either all real or all lie on the unit circle in the complex plane, a quantitative normal approximation bound for $X_n$ is established in a unified way. In the real rooted case the result is classical and only involves the variances of $X_n$, while in the cyclotomic case the fourth cumulants or moments of $X_n$ appear in addition. The proofs are elementary and based on the Stein-Tikhomirov method.

math.PR↗
arXiv2023

Probabilistic Limit Theorems Induced by the Zeros of Polynomials

Sequences of discrete random variables are studied whose probability generating functions are zero-free in a sector of the complex plane around the positive real axis. Sharp bounds on the cumulants of all orders are stated, leading to Berry-Esseen bounds, moderate deviation results, concentration inequalities and mod-Gaussian convergence. In addition, an alternate proof of the cumulant bound with improved constants for a class of polynomials all of whose roots lie on the unit circle is provided. A variety of examples is discussed in detail.

math.PR↗
arXiv2023

Vertex number of the typical cell in a tri-directional Poisson line tessellation

This paper deals with the typical cell in a Poisson line tessellation in the plane whose directional distribution is concentrated on three equally spread values with possibly different weights. Such a random polygon can only be a triangle, a quadrilateral, a pentagon or a hexagon. The probability for each of these cases is determined explicitly in terms of the weights. Extremal cases are discussed as well.

math.PR↗
arXiv2023

The volume of random simplices from elliptical distributions in high dimension

Random simplices and more general random convex bodies of dimension $p$ in $\mathbb{R}^n$ with $p\leq n$ are considered, which are generated by random vectors having an elliptical distribution. In the high-dimensional regime, that is, if $p\to\infty$ and $n\to\infty$ in such a way that $p/n\toγ\in(0,1)$, a central and a stable limit theorem for the logarithmic volume of random simplices and random convex bodies is shown. The result follows from a related central limit theorem for the log-determinant of $p\times n$ random matrices whose rows are copies of a random vector with an elliptical distribution, which is established as well.

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arXiv2023

Large deviations for hyperbolic $k$-nearest neighbor balls

We prove a large deviation principle for the point process of large Poisson $k$-nearest neighbor balls in hyperbolic space. More precisely, we consider a stationary Poisson point process of unit intensity in a growing sampling window in hyperbolic space. We further take a growing sequence of thresholds such that there is a diverging expected number of Poisson points whose $k$-nearest neighbor ball has a volume exceeding this threshold. Then, the point process of exceedances satisfies a large deviation principle whose rate function is described in terms of a relative entropy. The proof relies on a fine coarse-graining technique such that inside the resulting blocks the exceedances are approximated by independent Poisson point processes.

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arXiv2023

Intersections of Poisson $ k $-flats in constant curvature spaces

Poisson processes in the space of $k$-dimensional totally geodesic subspaces ($k$-flats) in a $d$-dimensional standard space of constant curvature $κ\in\{-1,0,1\}$ are studied, whose distributions are invariant under the isometries of the space. We consider the intersection processes of order $m$ together with their $(d-m(d-k))$-dimensional Hausdorff measure within a geodesic ball of radius $r$. Asymptotic normality for fixed $r$ is shown as the intensity of the underlying Poisson process tends to infinity for all $m$ satisfying $d-m(d-k)\geq 0$. For $κ\in\{-1,0\}$ the problem is also approached in the set-up where the intensity is fixed and $r$ tends to infinity. Again, if $2k\le d+1$ a central limit theorem is shown for all possible values of $m$. However, while for $κ=0$ asymptotic normality still holds if $2k>d+1$, we prove for $κ=-1$ convergence to a non-Gaussian infinitely divisible limit distribution in the special case $m=1$. The proof of asymptotic normality is based on the analysis of variances and general bounds available from the Malliavin--Stein method. We also show for general $κ\in\{-1,0,1\}$ that, roughly speaking, the variances within a general observation window $W$ are maximal if and only if $W$ is a geodesic ball having the same volume as $W$. Along the way we derive a new integral-geometric formula of Blaschke--Petkantschin type in a standard space of constant curvature.

math.PR↗
arXiv2022

The proportion of triangles in a class of anisotropic Poisson line tessellations

Stationary Poisson processes of lines in the plane are studied whose directional distributions are concentrated on $k \ge 3$ equally spread directions. The random lines of such processes decompose the plane into a collection of random polygons, which form a so-called Poisson line tessellation. The focus of this paper is to determine the proportion of triangles in such tessellations, or equivalently, the probability that the typical cell is a triangle. As a by-product, a new deviation of Miles' classical result for the isotropic case is obtained by an approximation argument.

math.PR↗
arXiv2022

Second-order properties for planar Mondrian tessellations

In this paper planar STIT tesselations with weighted axis-parallel cutting directions are considered. They are known also as weighted planar Mondrian tesselations in the machine learning literature, where they are used in random forest learning and kernel methods. Various second-order properties of such random tessellations are derived, in particular, explicit formulas are obtained for suitably adapted versions of the pair- and cross-correlation functions of the length measure on the edge skeleton and the vertex point process. Also, explicit formulas and the asymptotic behaviour of variances are discussed in detail.

math.PR↗
arXiv2022

Variance expansion and Berry-Esseen bound for the number of vertices of a random polygon in a polygon

Fix a container polygon $P$ in the plane and consider the convex hull $P_n$ of $n\geq 3$ independent and uniformly distributed in $P$ random points. In the focus of this paper is the vertex number of the random polygon $P_n$. The precise variance expansion for the vertex number is determined up to the constant-order term, a result which can be considered as a second-order analogue of the classical expansion for the expectation of Rényi and Sulanke (1963). Moreover, a sharp Berry-Esseen bound is derived for the vertex number of the random polygon $P_n$, which is of the same order as the square-root of the variance. The main idea behind the proof of both results is a decomposition of the boundary of the random polygon $P_n$ into random convex chains and a careful merging of the variance expansions and Berry-Esseen bounds for the vertex numbers of the individual chains.

math.PR↗
arXiv2022

Beta-star polytopes and hyperbolic stochastic geometry

Motivated by problems of hyperbolic stochastic geometry we introduce and study the class of beta-star polytopes. A beta-star polytope is defined as the convex hull of an inhomogeneous Poisson processes on the complement of the unit ball in $\mathbb{R}^d$ with density proportional to $(||x||^2-1)^β$, where $||x|| > 1$ and $β>d/2$. Explicit formulas for various geometric and combinatorial functionals associated with beta-star polytopes are provided, including the expected number of $k$-dimensional faces, the expected external angle sums and the expected intrinsic volumes. Beta-star polytopes are relevant in the context of hyperbolic stochastic geometry, since they are tightly connected to the typical cell of a Poisson-Voronoi tessellation as well as the zero cell of a Poisson hyperplane tessellation in hyperbolic space. The general results for beta-star polytopes are used to provide explicit formulas for the expected $f$- vector of the typical hyperbolic Poisson-Voronoi cell and the hyperbolic Poisson zero cell. Their asymptotics for large intensities and their monotonicity behaviour is discussed as well. Finally, stochastic geometry in the de Sitter half-space is studied as the hyperbolic analogue to recent investigations about random cones generated by random points on half-spheres in spherical or conical stochastic geometry.

math.PR↗
arXiv2022

Spherical convex hull of random points on a wedge

Consider two half-spaces $H_1^+$ and $H_2^+$ in $\mathbb{R}^{d+1}$ whose bounding hyperplanes $H_1$ and $H_2$ are orthogonal and pass through the origin. The intersection $\mathbb{S}_{2,+}^d:=\mathbb{S}^d\cap H_1^+\cap H_2^+$ is a spherical convex subset of the $d$-dimensional unit sphere $\mathbb{S}^d$, which contains a great subsphere of dimension $d-2$ and is called a spherical wedge. Choose $n$ independent random points uniformly at random on $\mathbb{S}_{2,+}^d$ and consider the expected facet number of the spherical convex hull of these points. It is shown that, up to terms of lower order, this expectation grows like a constant multiple of $\log n$. A similar behaviour is obtained for the expected facet number of a homogeneous Poisson point process on $\mathbb{S}_{2,+}^d$. The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a half-sphere.

math.PR↗
arXiv2022

Large deviations for uniform projections of $p$-radial distributions on $\ell_p^n$-balls

We consider products of uniform random variables from the Stiefel manifold of orthonormal $k$-frames in $\mathbb{R}^n$, $k \le n$, and random vectors from the $n$-dimensional $\ell_p^n$-ball $\mathbb{B}_p^n$ with certain $p$-radial distributions, $p\in[1,\infty)$. The distribution of this product geometrically corresponds to the projection of the $p$-radial distribution on $\mathbb{B}^n_p$ onto a random $k$-dimensional subspace. We derive large deviation principles (LDPs) on the space of probability measures on $\mathbb{R}^k$ for sequences of such projections.

math.PR↗
arXiv2022

Approaching the coupon collector's problem with group drawings via Stein's method

In this paper the coupon collector's problem with group drawings is studied. Assume there are $ n $ different coupons. At each time precisely $ s $ of the $ n $ coupons are drawn, where all choices are supposed to have equal probability. The focus lies on the fluctuations, as $n\to\infty$, of the number $Z_{n,s}(k_n)$ of coupons that have not been drawn in the first $k_n$ drawings. Using a size-biased coupling construction together with Stein's method for normal approximation, a quantitative central limit theorem for $Z_{n,s}(k_n)$ is shown for the case that $k_n={n\over s}(α\log(n)+x)$, where $0<α<1$ and $x\in\mathbb{R}$. The same coupling construction is used to retrieve a quantitative Poisson limit theorem in the boundary case $α=1$, again using Stein's method.

math.PR↗
arXiv2021

Sharpening the probabilistic Arithmetic-Geometric Mean Inequality

We consider the $p$-generalized arithmetic-geometric mean inequality for vectors chosen randomly from the $\ell_p^n$-ball in $\mathbb{R}^n$. In this setting the inequality can be improved or reversed up to a respective scalar constant with high probability, and central limit theorems and large deviation results with respect to this constant have been shown. We sharpen these large deviation results in the spirit of Bahadur and Ranga Rao, thereby providing concrete and asymptotically exact estimates on a non-logarithmic scale for the probability of the inequality being improvable or reversible up to a constant, respectively.

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