arXiv subjects

Christoph Thaele

Publications and source records attributed to Christoph Thaele.

At least 55 records · Page 3Linked to original sources

arXiv2017

Multivariate central limit theorems for Rademacher functionals with applications

Quantitative multivariate central limit theorems for general functionals of possibly non-symmetric and non-homogeneous infinite Rademacher sequences are proved by combining discrete Malliavin calculus with the smart path method for normal approximation. In particular, a discrete multivariate second-order Poincaré inequality is developed. As a first application, the normal approximation of vectors of subgraph counting statistics in the Erdős-Rényi random graph is considered. In this context, we further specialize to the normal approximation of vectors of vertex degrees. In a second application we prove a quantitative multivariate central limit theorem for vectors of intrinsic volumes induced by random cubical complexes.

math.PR↗

arXiv2017

Gaussian fluctuations for edge counts in high-dimensional random geometric graphs

Consider a stationary Poisson point process in $\mathbb{R}^d$ and connect any two points whenever their distance is less than or equal to a prescribed distance parameter. This construction gives rise to the well known random geometric graph. The number of edges of this graph is counted that have midpoint in the $d$-dimensional unit ball. A quantitative central limit theorem for this counting statistic is derived, as the space dimension $d$ and the intensity of the Poisson point process tend to infinity simultaneously.

math.PR↗

arXiv2016

A Mecke-type formula and Markov properties for STIT tessellation processes

An analogue of the classical Mecke formula for Poisson point processes is proved for the class of space-time STIT tessellation processes. From this key identity the Markov property of a class of associated random processes is derived. This in turn is used to determine the distribution of the number of internal vertices of the typical maximal tessellation segment.

math.PR↗

arXiv2016

A new quantitative central limit theorem on the Wiener space with applications to Gaussian processes

Quantitative limit theorems for non-linear functionals on the Wiener space are considered. Given the possibly infinite sequence of kernels of the chaos decomposition of such a functional, an estimate for different probability distances between the functional and a Gaussian random variable in terms of contraction norms of these kernels is derived. The applicability of this result is demonstrated by means of the Breuer-Major theorem, unfolding thereby a new connection between the Hermite rank of the considered function and a chaotic gap. Especially, power variations of the fractional Brownian motion and processes belonging to the Cauchy class are studied.

math.PR↗

arXiv2016

Limit theory for the Gilbert graph

For a given homogeneous Poisson point process in $\mathbb{R}^d$ two points are connected by an edge if their distance is bounded by a prescribed distance parameter. The behaviour of the resulting random graph, the Gilbert graph or random geometric graph, is investigated as the intensity of the Poisson point process is increased and the distance parameter goes to zero. The asymptotic expectation and covariance structure of a class of length-power functionals are computed. Distributional limit theorems are derived that have a Gaussian, a stable or a compound Poisson limiting distribution. Finally, concentration inequalities are provided using a concentration inequality for the convex distance.

math.PR↗

arXiv2015

Concentration and moderate deviations for Poisson polytopes and polyhedra

The convex hull generated by the restriction to the unit ball of a stationary Poisson point process in the $d$-dimensional Euclidean space is considered. By establishing sharp bounds on cumulants, exponential estimates for large deviation probabilities are derived and the relative error in the central limit theorem on a logarithmic scale is investigated for a large class of key geometric characteristics. This includes the number of lower-dimensional faces and the intrinsic volumes of the random polytopes. Furthermore, moderate deviation principles for the spatial empirical measures induced by these functionals are also established using the method of cumulants. The results are applied to deduce, by duality, fine probabilistic estimates and moderate deviation principles for combinatorial parameters of a class of zero cells associated with Poisson hyperplane mosaics. As a special case this comprises the typical Poisson-Voronoi cell conditioned on having large inradius.

math.PR↗

arXiv2015

A random cell splitting scheme on the sphere

A random recursive cell splitting scheme of the $2$-dimensional unit sphere is considered, which is the spherical analogue of the STIT tessellation process from Euclidean stochastic geometry. First-order moments are computed for a large array of combinatorial and metric parameters of the induced splitting tessellations by means of martingale methods combined with tools from spherical integral geometry. The findings are compared with those in the Euclidean case, making thereby transparent the influence of the curvature of the underlying space. Moreover, the capacity functional is computed and the point process that arises from the intersection of a splitting tessellation with a fixed great circle is characterized.

math.PR↗

arXiv2015

Discrete Malliavin-Stein method: Berry-Esseen bounds for random graphs and percolation

A new Berry-Esseen bound for non-linear functionals of non-symmetric and non-homogeneous infinite Rademacher sequences is established. It is based on a discrete version of the Malliavin-Stein method and an analysis of the discrete Ornstein-Uhlenbeck semigroup. The result is applied to sub-graph counts and to the number of vertices having a prescribed degree in the Erdős-Renyi random graph. A further application deals with a percolation problem on trees.

math.PR↗

arXiv2015

Poisson point process convergence and extreme values in stochastic geometry

Let $η_t$ be a Poisson point process with intensity measure $tμ$, $t>0$, over a Borel space $\mathbb{X}$, where $μ$ is a fixed measure. Another point process $ξ_t$ on the real line is constructed by applying a symmetric function $f$ to every $k$-tuple of distinct points of $η_t$. It is shown that $ξ_t$ behaves after appropriate rescaling like a Poisson point process, as $t\to\infty$, under suitable conditions on $η_t$ and $f$. This also implies Weibull limit theorems for related extreme values. The result is then applied to investigate problems arising in stochastic geometry, including small cells in Voronoi tessellations, random simplices generated by non-stationary hyperplane processes, triangular counts with angular constraints and non-intersecting $k$-flats. Similar results are derived if the underlying Poisson point process is replaced by a binomial point process.

math.PR↗

arXiv2015

Poisson polyhedra in high dimensions

The zero cell of a parametric class of random hyperplane tessellations depending on a distance exponent and an intensity parameter is investigated, as the space dimension tends to infinity. The model includes the zero cell of stationary and isotropic Poisson hyperplane tessellations as well as the typical cell of a stationary Poisson Voronoi tessellation as special cases. It is shown that asymptotically in the space dimension, with overwhelming probability these cells satisfy the hyperplane conjecture, if the distance exponent and the intensity parameter are suitably chosen dimension-dependent functions. Also the high dimensional limits of the mean number of faces are explored and the asymptotic behaviour of an isoperimetric ratio is analysed. In the background are new identities linking the $f$-vector of the zero cell to certain dual intrinsic volumes.

math.MG↗

arXiv2015

A four moments theorem for Gamma limits on a Poisson chaos

This paper deals with sequences of random variables belonging to a fixed chaos of order $q$ generated by a Poisson random measure on a Polish space. The problem is investigated whether convergence of the third and fourth moment of such a suitably normalized sequence to the third and fourth moment of a centred Gamma law implies convergence in distribution of the involved random variables. A positive answer is obtained for $q=2$ and $q=4$. The proof of this four moments theorem is based on a number of new estimates for contraction norms. Applications concern homogeneous sums and $U$-statistics on the Poisson space.

math.PR↗

arXiv2014

On the arrangement of cells in planar STIT and Poisson line tessellations

It is well known that the distributions of the interiors of the typical cell of a Poisson line tessellation and a STIT tessellation with the same parameters coincide. In this paper, differences in the arrangement of the cells in these two tessellation models are investigated. In particular, characteristics of the set of cells neighbouring the typical cell are studied. Furthermore, the pair-correlation function and several mark correlation functions of the point processes of cell centres are estimated and compared.

math.PR↗

arXiv2014

Limit theorems for iteration stable tessellations

The intent of this paper is to describe the large scale asymptotic geometry of iteration stable (STIT) tessellations in $\mathbb{R}^d$, which form a rather new, rich and flexible class of random tessellations considered in stochastic geometry. For this purpose, martingale tools are combined with second-order formulas proved earlier to establish limit theorems for STIT tessellations. More precisely, a Gaussian functional central limit theorem for the surface increment process induced a by STIT tessellation relative to an initial time moment is shown. As second main result, a central limit theorem for the total edge length/facet surface is obtained, with a normal limit distribution in the planar case and, most interestingly, with a nonnormal limit showing up in all higher space dimensions.

math.PR↗

arXiv2014

Geometry of iteration stable tessellations: Connection with Poisson hyperplanes

Since the seminal work by Nagel and Weiss, the iteration stable (STIT) tessellations have attracted considerable interest in stochastic geometry as a natural and flexible, yet analytically tractable model for hierarchical spatial cell-splitting and crack-formation processes. We provide in this paper a fundamental link between typical characteristics of STIT tessellations and those of suitable mixtures of Poisson hyperplane tessellations using martingale techniques and general theory of piecewise deterministic Markov processes (PDMPs). As applications, new mean values and new distributional results for the STIT model are obtained.

math.PR↗

arXiv2014

Central limit theorems for the radial spanning tree

Consider a homogeneous Poisson point process in a compact convex set in $d$-dimensional Euclidean space which has interior points and contains the origin. The radial spanning tree is constructed by connecting each point of the Poisson point process with its nearest neighbour that is closer to the origin. For increasing intensity of the underlying Poisson point process the paper provides expectation and variance asymptotics as well as central limit theorems with rates of convergence for a class of edge functionals including the total edge length.

math.PR↗

arXiv2014

Intersection and proximity of processes of flats

Weakly stationary random processes of $k$-dimensional affine subspaces (flats) in $\mathbb{R}^n$ are considered. If $2k\geq n$, then intersection processes are investigated, while in the complementary case $2k<n$ a proximity process is introduced. The intensity measures of these processes are described in terms of parameters of the underlying $k$-flat process. By a translation into geometric parameters of associated zonoids and by means of integral transformations, several new uniqueness and stability results for these processes of flats are derived. They rely on a combination of known and novel estimates for area measures of zonoids, which are also developed in the paper. Finally, an asymptotic second-order analysis as well as central and non-central limit theorems for length-power direction functionals of proximity processes derived from stationary Poisson $k$-flat process complement earlier works for intersection processes.

math.MG↗

arXiv2014

Cumulants on Wiener chaos: moderate deviations and the fourth moment theorem

A moderate deviation principle as well as moderate and large deviation inequalities for a sequence of elements living inside a fixed Wiener chaos associated with an isonormal Gaussian process are shown. The conditions under which the results are derived coincide with those of the celebrated fourth moment theorem of Nualart and Peccati. The proofs rely on sharp estimates for cumulants. As applications, explosive integrals of a Brownian sheet, a discretized version of the quadratic variation of a fractional Brownian motion and the sample bispectrum of a spherical Gaussian random field are considered.

math.PR↗

arXiv2014

Berry-Esseen bounds and multivariate limit theorems for functionals of Rademacher sequences

Berry-Esseen bounds for non-linear functionals of infinite Rademacher sequences are derived by means of the Malliavin-Stein method. Moreover, multivariate extensions for vectors of Rademacher functionals are shown. The results establish a connection to small-ball probabilities and shed new light onto the relation between central limit theorems on the Rademacher chaos and norms of contraction operators. Applications concern infinite weighted 2-runs, a combinatorial central limit theorem and traces of Bernoulli random matrices.

math.PR↗