arXiv subjects

Christoph Thäle

Publications and source records attributed to Christoph Thäle.

At least 55 records · Page 3Linked to original sources

arXiv2021

Multi-Dimensional Normal Approximation of Heavy-Tailed Moving Averages

In this paper we extend the refined second-order Poincaré inequality for Poisson functionals from a one-dimensional to a multi-dimensional setting. Its proof is based on a multivariate version of the Malliavin-Stein method for normal approximation on Poisson spaces. We also present an application to partial sums of vector-valued functionals of heavy-tailed moving averages. The extension allows a functional with multivariate arguments, i.e. multiple moving averages and also multivariate values of the functional. Such a set-up has previously not been explored in the framework of stable moving average processes. It can potentially capture probabilistic properties which cannot be described solely by the one-dimensional marginals, but instead require the joint distribution.

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arXiv2021

The $β$-Delaunay tessellation I: Description of the model and geometry of typical cells

In this paper two new classes of stationary random simplicial tessellations, the so-called $β$- and $β'$-Delaunay tessellations, are introduced. Their construction is based on a space-time paraboloid hull process and generalizes that of the classical Poisson-Delaunay tessellation. The distribution of volume-power weighted typical cells is explicitly identified, establishing thereby a remarkable connection to the classes of $β$- and $β'$-polytopes. These representations are used to determine principal characteristics of such cells, including volume moments, expected angle sums and cell intensities.

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arXiv2021

Variance asymptotics and central limit theory for geometric functionals of Poisson cylinder processes

This paper deals with the union set of a stationary Poisson process of cylinders in $\mathbb{R}^n$ having an $(n-m)$-dimensional base and an $m$-dimensional direction space, where $m\in\{0,1,\ldots,n-1\}$ and $n\geq 2$. The concept simultaneously generalises those of a Boolean model and a Poisson hyperplane or $m$-flat process. Under very general conditions on the typical cylinder base a Berry-Esseen bound for the volume of the union set within a sequence of growing test sets is derived. Assuming convexity of the cylinder bases and of the window a similar result is shown for a broad class of geometric functionals, including the intrinsic volumes. In this context the asymptotic variance constant is analysed in detail, which in contrast to the Boolean model leads to a new degeneracy phenomenon. A quantitative central limit theory is developed in a multivariate set-up as well.

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arXiv2021

The $β$-Delaunay tessellation IV: Mixing properties and central limit theorems

Various mixing properties of $β$-, $β'$- and Gaussian Delaunay tessellations in $\mathbb{R}^{d-1}$ are studied. It is shown that these tessellation models are absolutely regular, or $β$-mixing. In the $β$- and the Gaussian case exponential bounds for the absolute regularity coefficients are found. In the $β'$-case these coefficients show a polynomial decay only. In the background are new and strong concentration bounds on the radius of stabilization of the underlying construction. Using a general device for absolutely regular stationary random tessellations, central limit theorems for a number of geometric parameters of $β$- and Gaussian Delaunay tessellations are established. This includes the number of $k$-dimensional faces and the $k$-volume of the $k$-sk

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arXiv2021

A simplified second-order Gaussian Poincaré inequality in discrete setting with applications

In this paper, a simplified second-order Gaussian Poincaré inequality for normal approximation of functionals over infinitely many Rademacher random variables is derived. It is based on a new bound for the Kolmogorov distance between a general Rademacher functional and a Gaussian random variable, which is established by means of the discrete Malliavin-Stein method and is of independent interest. As an application, the number of vertices with prescribed degree and the subgraph counting statistic in the Erdös-Rényi random graph are discussed. The number of vertices of fixed degree is also studied for percolation on the Hamming hypercube. Moreover, the number of isolated faces in the Linial-Meshulam-Wallach random $κ$-complex and infinite weighted 2-runs are treated.

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arXiv2021

Optimal Variance--Gamma approximation on the second Wiener chaos

In this paper, we consider a target random variable $Y \sim \CVG$ distributed according to a centered Variance--Gamma distribution. For a generic random element $F=I_2(f)$ in the second Wiener chaos with $\E[F^2]= \E[Y^2]$ we establish a non-asymptotic optimal bound on the distance between $F$ and $Y$ in terms of the maximum of difference of the first six cumulants. This six moment theorem extends the celebrated optimal fourth moment theorem of I.\ Nourdin \& G.\ Peccati for normal approximation. The main body of our analysis constitutes a splitting technique for test functions in the Banach space of Lipschitz functions relying on the compactness of the Stein operator. The recent developments around Stein method for Variance--Gamma approximation by R.\ Gaunt play a significant role in our study. As an application we consider the generalized Rosenblatt process at the extreme critical exponent, first studied by S.\ Bai \& M.\ Taqqu.

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arXiv2021

Sharp inequalities for the mean distance of random points in convex bodies

For a convex body $K\subset\mathbb{R}^d$ the mean distance $Δ(K)=\mathbb{E}|X_1-X_2|$ is the expected Euclidean distance of two independent and uniformly distributed random points $X_1,X_2\in K$. Optimal lower and upper bounds for ratio between $Δ(K)$ and the first intrinsic volume $V_1(K)$ of $K$ (normalized mean width) are derived and degenerate extremal cases are discussed. The argument relies on Riesz's rearrangement inequality and the solution of an optimization problem for powers of concave functions. The relation with results known from the existing literature is reviewed in detail.

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arXiv2021

Random cones in high dimensions II: Weyl cones

We consider two models of random cones together with their duals. Let $Y_1,\dots,Y_n$ be independent and identically distributed random vectors in $\mathbb R^d$ whose distribution satisfies some mild condition. The random cones $G_{n,d}^A$ and $G_{n,d}^B$ are defined as the positive hulls $\text{pos}\{Y_1-Y_2,\dots,Y_{n-1}-Y_n\}$, respectively $\text{pos}\{Y_1-Y_2,\dots,Y_{n-1}-Y_n,Y_n\}$, conditioned on the event that the respective positive hull is not equal to $\mathbb R^d$. We prove limit theorems for various expected geometric functionals of these random cones, as $n$ and $d$ tend to infinity in a coordinated way. This includes limit theorems for the expected number of $k$-faces and the $k$-th conic quermassintegrals, as $n$, $d$ and sometimes also $k$ tend to infinity simultaneously. Moreover, we uncover a phase transition in high dimensions for the expected statistical dimension for both models of random cones.

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arXiv2021

The $β$-Delaunay tessellation III: Kendall's problem and limit theorems in high dimensions

The $β$-Delaunay tessellation in $\mathbb{R}^{d-1}$ is a generalization of the classical Poisson-Delaunay tessellation. As a first result of this paper we show that the shape of a weighted typical cell of a $β$-Delaunay tessellation, conditioned on having large volume, is close to the shape of a regular simplex in $\mathbb{R}^{d-1}$. This generalizes earlier results of Hug and Schneider about the typical (non-weighted) Poisson-Delaunay simplex. Second, the asymptotic behaviour of the volume of weighted typical cells in high-dimensional $β$-Delaunay tessellation is analysed, as $d\to\infty$. In particular, various high dimensional limit theorems, such as quantitative central limit theorems as well as moderate and large deviation principles, are derived.

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arXiv2021

The $β$-Delaunay tessellation II: The Gaussian limit tessellation

We study the weak convergence of $β$- and $β'$-Delaunay tessellations in $\mathbb{R}^{d-1}$ that were introduced in part I of this paper, as $β\to\infty$. The limiting stationary simplicial random tessellation, which is called the Gaussian-Delaunay tessellation, is characterized in terms of a space-time paraboloid hull process in $\mathbb{R}^{d-1}\times\mathbb{R}$. The latter object has previously appeared in the analysis of the number of shocks in the solution of the inviscid Burgers' equation and the description of the local asymptotic geometry of Gaussian random polytopes. In this paper it is used to define a new stationary random simplicial tessellation in $\mathbb{R}^{d-1}$. As for the $β$- and $β'$-Delaunay tessellation, the distribution of volume-power weighted typical cells in the Gaussian-Delaunay tessellation is explicitly identified, establishing thereby a new bridge to Gaussian random simplices. Also major geometric characteristics of these cells such as volume moments, expected angle sums and also the cell intensities of the Gaussian-Delaunay tessellation are investigated.

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arXiv2020

On random convex chains, orthogonal polynomials, PF sequences and probabilistic limit theorems

Let $T$ be the triangle in the plane with vertices $(0,0)$, $(0,1)$ and $(0,1)$. The convex hull of $(0,1)$, $(1,0)$ and $n$ independent random points uniformly distributed in $T$ is the random convex chain $T_n$. A three-term recursion for the probability generating function $G_n$ of the number $f_0(T_n)$ of vertices of $T_n$ is proved. Via the link to orthogonal polynomials it is shown that $G_n$ has precisely $n$ distinct real roots in $(-\infty,0]$ and that the sequence $p_k^{(n)}:=\mathbb{P}(f_0(T_n)=k)$, $k=1,\ldots,n$, is a Polya frequency (PF) sequence. A selection of probabilistic consequences of this surprising and remarkable fact are discussed in detail.

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arXiv2020

Weak convergence of the intersection point process of Poisson hyperplanes

This paper deals with the intersection point process of a stationary and isotropic Poisson hyperplane process in $\mathbb{R}^d$ of intensity $t>0$, where only hyperplanes that intersect a centred ball of radius $R>0$ are considered. Taking $R=t^{-\frac{d}{d+1}}$ it is shown that this point process converges in distribution, as $t\to\infty$, to a Poisson point process on $\mathbb{R}^d\setminus\{0\}$ whose intensity measure has power-law density proportional to $\|x\|^{-(d+1)}$ with respect to the Lebesgue measure. A bound on the speed of convergence in terms of the Kantorovich-Rubinstein distance is provided as well. In the background is a general functional Poisson approximation theorem on abstract Poisson spaces. Implications on the weak convergence of the convex hull of the intersection point process and the convergence of its $f$-vector are also discussed, disproving and correcting thereby a conjecture of Devroye and Toussaint [J.\ Algorithms 14.3 (1993), 381--394] in computational geometry.

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arXiv2020

Random inscribed polytopes in projective geometries

We establish central limit theorems for natural volumes of random inscribed polytopes in projective Riemannian or Finsler geometries. In addition, normal approximation of dual volumes and the mean width of random polyhedral sets are obtained. We deduce these results by proving a general central limit theorem for the weighted volume of the convex hull of random points chosen from the boundary of a smooth convex body according to a positive and continuous density in Euclidean space. In the background are geometric estimates for weighted surface bodies and Berry-Esseen bounds for functionals of independent random variables.

math.MG↗

arXiv2020

Faces in random great hypersphere tessellations

The concept of typical and weighted typical spherical faces for tessellations of the $d$-dimensional unit sphere, generated by $n$ independent random great hyperspheres distributed according to a non-degenerate directional distribution, is introduced and studied. Probabilistic interpretations for such spherical faces are given and their directional distributions are determined. Explicit formulas for the expected $f$-vector, the expected spherical Quermaßintegrals and the expected spherical intrinsic volumes are found in the isotropic case. Their limiting behaviour as $n\to\infty$ is discussed and compared to the corresponding notions and results in the Euclidean case. The expected statistical dimension and a problem related to intersection probabilities of spherical random polytopes is investigated.

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arXiv2020

A new approach to weak convergence of random cones and polytopes

A new approach to prove weak convergence of random polytopes on the space of compact convex sets is presented. This is used to show that the profile of the rescaled Schläfli random cone of a random conical tessellation generated by $n$ independent and uniformly distributed random linear hyperplanes in $\mathbb{R}^{d+1}$ weakly converges to the typical cell of a stationary and isotropic Poisson hyperplane tessellation in $\mathbb{R}^d$, as $n \to \infty$.

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arXiv2019

Asymptotic normality for random polytopes in non-Euclidean geometries

Asymptotic normality for the natural volume measure of random polytopes generated by random points distributed uniformly in a convex body in spherical or hyperbolic spaces is proved. Also the case of Hilbert geometries is treated and central limit theorems in Lutwak's dual Brunn--Minkowski theory are established. The results follow from a central limit theorem for weighted random polytopes in Euclidean spaces. In the background are Stein's method for normal approximation and geometric properties of weighted floating bodies.

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arXiv2019

Asymptotic normality for random simplices and convex bodies in high dimensions

Central limit theorems for the log-volume of a class of random convex bodies in $\mathbb{R}^n$ are obtained in the high-dimensional regime, that is, as $n\to\infty$. In particular, the case of random simplices pinned at the origin and simplices where all vertices are generated at random is investigated. The coordinates of the generating vectors are assumed to be independent and identically distributed with subexponential tails. In addition, asymptotic normality is established also for random convex bodies (including random simplices pinned at the origin) when the spanning vectors are distributed according to a radially symmetric probability measure on the $n$-dimensional $\ell_p$-ball. In particular, this includes the cone and the uniform probability measure.

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arXiv2019

A Berry-Esseén theorem for partial sums of functionals of heavy-tailed moving averages

In this paper we obtain Berry-Esseén bounds on partial sums of functionals of heavy-tailed moving averages, including the linear fractional stable noise, stable fractional ARIMA processes and stable Ornstein-Uhlenbeck processes. Our rates are obtained for the Wasserstein and Kolmogorov distances, and depend strongly on the interplay between the memory of the process, which is controlled by a parameter $α$, and its tail-index, which is controlled by a parameter $β$. In fact, we obtain the classical $\sqrt{1/ n}$ rate of convergence when the tails are not too heavy and the memory is not too strong, more precisely, when $αβ> 3$ or $αβ> 4$ in the case of Wasserstein and Kolmogorov distance, respectively. Our quantitative bounds rely on a new second-order Poincare inequality on the Poisson space, which we derive through a combination of Stein's method and Malliavin calculus. This inequality improves and generalizes a result by Last, Peccati, Schulte [Probab. Theory Relat. Fields 165 (2016)].

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