arXiv subjects

J. E. Yukich

Publications and source records attributed to J. E. Yukich.

At least 19 recordsLinked to original sources

arXiv2026

Quantitative CLTs for Geometric Statistics of Dependent Marked Point Processes

Given a geometric statistic expressible as a sum of scores which depend on local data, \citet{BYY19} established central limit theorems for centered and normalized versions of these statistics, subject to the underlying point process having fast decay of correlations and also subject to a variance growth condition. Building on this, \citet{CX23} derived rates of normal approximation, as measured by the Wasserstein distance, for statistics of point processes exhibiting fast decay of dependence. Here we go further and establish weak mixing conditions yielding rates of normal convergence for geometric statistics of marked point processes. The mixing conditions, which are in terms of verifiable geometric criteria, provide rates of normal approximation in the Wasserstein distance for statistics of a large class of point processes with dependent marks. Examples include statistics of determinantal point processes, unevenly spaced time series, continuum percolation, interacting diffusions on spatial random graphs, as well as local $U$-statistics.

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arXiv2026

Limit theory for Lipschitz-localized statistics in random geometric models

We study sums of locally dependent scores associated with general marked (i.e., labeled) Euclidean point processes. We introduce geometric mixing conditions on the underlying point process and a Lipschitz-"localization" condition on the scores, which jointly ensure a central limit theorem for the sums of the scores as well as expectation and variance asymptotics. Our localization condition is formulated using the bounded Lipschitz metric, providing a distributional criterion. This stands in contrast to the classical stabilization conditions in stochastic geometry, which are typically based on stopping-set constructions. To demonstrate the applicability of our general framework, we consider several stochastic processes indexed by spatial random graphs. These include spin systems, interacting diffusions, and interacting particle systems. In particular, spin systems highlight the importance of our localization condition. Additional applications include empirical random fields and geostatistical Boolean models.

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arXiv2026

Second-order Poincaré inequalities and localization on the Poisson space

Given a mean zero functional $F$ of a Poisson measure on a metric space, we apply the Malliavin-Stein method to establish sharpened second-order Poincaré inequalities for $F/\sqrt{\operatorname{Var} (F)}$ in terms of fourth moments of difference operators. The rates of normal approximation are expressed in the Kolmogorov and Wasserstein distances and require fewer error terms than corresponding previous results. When $F$ is expressible as a sum of score functions which are distributionally close to scores having short-range structure, then we deduce that $F/\sqrt{\operatorname{Var}(F)}$ satisfies Berry-Esseen bounds. The normal approximation criteria of the scores, here called bounded Lipschitz localization, are more general than stabilization criteria and allow for unbounded interactions of scores. This approach yields Berry-Esseen bounds for local U-statistics on metric measures spaces, localizing functionals on hyperbolic space, as well as for Poisson functionals in a space-time setting, with infinite time horizon, including statistics of spatial birth-growth models and Laguerre tessellations.

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arXiv2021

Rates of multivariate normal approximation for statistics in geometric probability

We employ stabilization methods and second order Poincaré inequalities to establish rates of multivariate normal convergence for a large class of vectors $(H_s^{(1)},...,H_s^{(m)})$, $s \geq 1$, of statistics of marked Poisson processes on $\mathbb{R}^d$, $d \geq 2$, as the intensity parameter $s$ tends to infinity. Our results are applicable whenever the constituent functionals $H_s^{(i)}$, $i\in\{1,...,m\}$, are expressible as sums of exponentially stabilizing score functions satisfying a moment condition. The rates are for the $d_2$-, $d_3$-, and $d_{convex}$-distances. When we compare with a centered Gaussian random vector, whose covariance matrix is given by the asymptotic covariances, the rates are in general unimprovable and are governed by the rate of convergence of $s^{-1} {\rm Cov}( H_s^{(i)}, H_s^{(j)})$, $i,j\in\{1,...,m\}$, to the limiting covariance, shown to be of order $s^{-1/d}$. We use the general results to deduce rates of multivariate normal convergence for statistics arising in random graphs and topological data analysis as well as for multivariate statistics used to test equality of distributions. Some of our results hold for stabilizing functionals of Poisson input on suitable metric spaces.

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arXiv2019

Convex hulls of perturbed random point sets

We consider the convex hull of the perturbed point process comprised of $n$ i.i.d. points, each distributed as the sum of a uniform point on the unit sphere $§^{d-1}$ and a uniform point in the $d$-dimensional ball centered at the origin and of radius $n^α, α\in (-\infty, \infty)$. This model, inspired by the smoothed complexity analysis introduced in computational geometry \cite{DGGT,ST}, is a perturbation of the classical random polytope. We show that the perturbed point process, after rescaling, converges in the scaling limit to one of five Poisson point processes according to whether $α$ belongs to one of five regimes. The intensity measure of the limit Poisson point process undergoes a transition at the values $α= \frac{-2} {d -1}$ and $α= \frac{2} {d + 1}$ and it gives rise to four rescalings for the $k$-face functional on perturbed data. These rescalings are used to establish explicit expectation asymptotics for the number of $k$-dimensional faces of the convex hull of either perturbed binomial or Poisson data. In the case of Poisson input, we establish explicit variance asymptotics and a central limit theorem for the number of $k$-dimensional faces. Finally it is shown that the rescaled boundary of the convex hull of the perturbed point process converges to the boundary of a parabolic hull process.

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arXiv2019

Multivariate second order Poincaré inequalities for Poisson functionals

Given a vector $F=(F_1,\dots,F_m)$ of Poisson functionals $F_1,\dots,F_m$, we investigate the proximity between $F$ and an $m$-dimensional centered Gaussian random vector $N_Σ$ with covariance matrix $Σ\in\mathbb{R}^{m\times m}$. Apart from finding proximity bounds for the $d_2$- and $d_3$-distances, based on classes of smooth test functions, we obtain proximity bounds for the $d_{convex}$-distance, based on the less tractable test functions comprised of indicators of convex sets. The bounds for all three distances are shown to be of the same order, which is presumably optimal. The bounds are multivariate counterparts of the univariate second order Poincaré inequalities and, as such, are expressed in terms of integrated moments of first and second order difference operators. The derived second order Poincaré inequalities for indicators of convex sets are made possible by a new bound on the second derivatives of the solution to the Stein equation for the multivariate normal distribution. We present applications to the multivariate normal approximation of first order Poisson integrals and of statistics of Boolean models.

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arXiv2018

Limit theory for geometric statistics of point processes having fast decay of correlations

Let $P$ be a simple,stationary point process having fast decay of correlations, i.e., its correlation functions factorize up to an additive error decaying faster than any power of the separation distance. Let $P_n:= P \cap W_n$ be its restriction to windows $W_n:= [-{1 \over 2}n^{1/d},{1 \over 2}n^{1/d}]^d \subset \mathbb{R}^d$. We consider the statistic $H_n^ξ:= \sum_{x \in P_n}ξ(x,P_n)$ where $ξ(x,P_n)$ denotes a score function representing the interaction of $x$ with respect to $P_n$. When $ξ$ depends on local data in the sense that its radius of stabilization has an exponential tail, we establish expectation asymptotics, variance asymptotics, and CLT for $H_n^ξ$ and, more generally, for statistics of the re-scaled, possibly signed, $ξ$-weighted point measures $μ_n^ξ := \sum_{x \in P_n} ξ(x,P_n) δ_{n^{-1/d}x}$, as $W_n \uparrow \mathbb{R}^d$. This gives the limit theory for non-linear geometric statistics (such as clique counts, intrinsic volumes of the Boolean model, and total edge length of the $k$-nearest neighbors graph) of $α$-determinantal point processes having fast decreasing kernels extending the CLTs of Soshnikov (2002) to non-linear statistics. It also gives the limit theory for geometric U-statistics of $α$-permanental point processes and the zero set of Gaussian entire functions, extending the CLTs of Nazarov and Sodin (2012) and Shirai and Takahashi (2003), which are also confined to linear statistics. The proof of the central limit theorem relies on a factorial moment expansion originating in Blaszczyszyn (1995), Blaszczyszyn, Merzbach, Schmidt (1997) to show the fast decay of the correlations of $ξ$-weighted point measures. The latter property is shown to imply a condition equivalent to Brillinger mixing and consequently yields the CLT for $μ_n^ξ$ via an extension of the cumulant method.

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arXiv2017

Normal approximation for stabilizing functionals

We establish presumably optimal rates of normal convergence with respect to the Kolmogorov distance for a large class of geometric functionals of marked Poisson and binomial point processes on general metric spaces. The rates are valid whenever the geometric functional is expressible as a sum of exponentially stabilizing score functions satisfying a moment condition. By incorporating stabilization methods into the Malliavin-Stein theory, we obtain rates of normal approximation for sums of stabilizing score functions which either improve upon existing rates or are the first of their kind. Our general rates hold for functionals of marked input on spaces more general than full-dimensional subsets of $\mathbb{R}^d$, including $m$-dimensional Riemannian manifolds, $m\leq d$. We use the general results to deduce improved and new rates of normal convergence for several functionals in stochastic geometry, including those whose variances re-scale as the volume or the surface area of an underlying set. In particular, we improve upon rates of normal convergence for the $k$-face and $i$th intrinsic volume functionals of the convex hull of Poisson and binomial random samples in a smooth convex body in dimension $d\geq 2$. We also provide improved rates of normal convergence for statistics of nearest neighbors graphs and high-dimensional data sets, the number of maximal points in a random sample, estimators of surface area and volume arising in set approximation via Voronoi tessellations, and clique counts in generalized random geometric graphs.

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arXiv2016

Asymptotic theory for statistics of the Poisson--Voronoi approximation

This paper establishes expectation and variance asymptotics for statistics of the Poisson--Voronoi approximation of general sets, as the underlying intensity of the Poisson point process tends to infinity. Statistics of interest include volume, surface area, Hausdorff measure, and the number of faces of lower-dimensional skeletons. We also consider the complexity of the so-called Voronoi zone and the iterated Voronoi approximation. Our results are consequences of general limit theorems proved with an abstract Steiner-type formula applicable in the setting of sums of stabilizing functionals.

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arXiv2016

Variance Asymptotics and Scaling Limits for Random Polytopes

Let K be a convex set in R d and let K $λ$ be the convex hull of a homogeneous Poisson point process P $λ$ of intensity $λ$ on K. When K is a simple polytope, we establish scaling limits as $λ$ $\rightarrow$ $\infty$ for the boundary of K $λ$ in a vicinity of a vertex of K and we give variance asymptotics for the volume and k-face functional of K $λ$, k $\in$ {0, 1, ..., d -- 1}, resolving an open question posed in [18]. The scaling limit of the boundary of K $λ$ and the variance asymptotics are described in terms of a germ-grain model consisting of cone-like grains pinned to the extreme points of a Poisson point process on R d--1 $\times$ R having intensity $\sqrt$ de dh dhdv.

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arXiv2015

Surface order scaling in stochastic geometry

Let $\mathcal{P}_λ:=\mathcal{P}_{λκ}$ denote a Poisson point process of intensity $λκ$ on $[0,1]^d,d\geq2$, with $κ$ a bounded density on $[0,1]^d$ and $λ\in(0,\infty)$. Given a closed subset $\mathcal{M}\subset[0,1]^d$ of Hausdorff dimension $(d-1)$, we consider general statistics $\sum_{x\in\mathcal{P}_λ}ξ(x,\mathcal{P} _λ,\mathcal{M})$, where the score function $ξ$ vanishes unless the input $x$ is close to $\mathcal{M}$ and where $ξ$ satisfies a weak spatial dependency condition. We give a rate of normal convergence for the rescaled statistics $\sum_{x\in\mathcal{ P}_λ}ξ(λ^{1/d}x,λ^{1/d}\mathcal{P}_λ,λ^{1/d}\mathcal{M})$ as $λ\to\infty$. When $\mathcal{M}$ is of class $C^2$, we obtain weak laws of large numbers and variance asymptotics for these statistics, showing that growth is surface order, that is, of order $\mathrm{Vol}(λ^{1/d}\mathcal{M})$. We use the general results to deduce variance asymptotics and central limit theorems for statistics arising in stochastic geometry, including Poisson-Voronoi volume and surface area estimators, answering questions in Heveling and Reitzner [Ann. Appl. Probab. 19 (2009) 719-736] and Reitzner, Spodarev and Zaporozhets [Adv. in Appl. Probab. 44 (2012) 938-953]. The general results also yield the limit theory for the number of maximal points in a sample.

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arXiv2014

Variance asymptotics and scaling limits for Gaussian Polytopes

Let $K_n$ be the convex hull of i.i.d. random variables distributed according to the standard normal distribution on $\R^d$. We establish variance asymptotics as $n \to \infty$ for the re-scaled intrinsic volumes and $k$-face functionals of $K_n$, $k \in \{0,1,...,d-1\}$, resolving an open problem. Variance asymptotics are given in terms of functionals of germ-grain models having parabolic grains with apices at a Poisson point process on $\R^{d-1} \times \R$ with intensity $e^h dh dv$. The scaling limit of the boundary of $K_n$ as $n \to \infty$ converges to a festoon of parabolic surfaces, coinciding with that featuring in the geometric construction of the zero viscosity solution to Burgers' equation with random input.

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arXiv2014

Normal approximation of Gibbsian sums in geometric probability

This paper concerns the asymptotic behavior of a random variable $W_λ$ resulting from the summation of the functionals of a Gibbsian spatial point process over windows $Q_λ\uparrow R^d$. We establish conditions ensuring that $W_λ$ has volume order fluctuations, that is they coincide with the fluctuations of functionals of Poisson spatial point processes. We combine this result with Stein's method to deduce rates of normal approximation for $W_λ$, as $λ\to\infty$. Our general results establish variance asymptotics and central limit theorems for statistics of random geometric and related Euclidean graphs on Gibbsian input. We also establish similar limit theory for claim sizes of insurance models with Gibbsian input, the number of maximal points of a Gibbsian sample, and the size of spatial birth-growth models with Gibbsian input.

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arXiv2013

Limit theory for point processes in manifolds

Let $Y_i,i\geq1$, be i.i.d. random variables having values in an $m$-dimensional manifold $\mathcal {M}\subset \mathbb{R}^d$ and consider sums $\sum_{i=1}^nξ(n^{1/m}Y_i,\{n^{1/m}Y_j\}_{j=1}^n)$, where $ξ$ is a real valued function defined on pairs $(y,\mathcal {Y})$, with $y\in \mathbb{R}^d$ and $\mathcal {Y}\subset \mathbb{R}^d$ locally finite. Subject to $ξ$ satisfying a weak spatial dependence and continuity condition, we show that such sums satisfy weak laws of large numbers, variance asymptotics and central limit theorems. We show that the limit behavior is controlled by the value of $ξ$ on homogeneous Poisson point processes on $m$-dimensional hyperplanes tangent to $\mathcal {M}$. We apply the general results to establish the limit theory of dimension and volume content estimators, Rényi and Shannon entropy estimators and clique counts in the Vietoris-Rips complex on $\{Y_i\}_{i=1}^n$.

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arXiv2013

Brownian limits, local limits and variance asymptotics for convex hulls in the ball

Schreiber and Yukich [Ann. Probab. 36 (2008) 363-396] establish an asymptotic representation for random convex polytope geometry in the unit ball $\mathbb{B}^d, d\geq2$, in terms of the general theory of stabilizing functionals of Poisson point processes as well as in terms of generalized paraboloid growth processes. This paper further exploits this connection, introducing also a dual object termed the paraboloid hull process. Via these growth processes we establish local functional limit theorems for the properly scaled radius-vector and support functions of convex polytopes generated by high-density Poisson samples. We show that direct methods lead to explicit asymptotic expressions for the fidis of the properly scaled radius-vector and support functions. Generalized paraboloid growth processes, coupled with general techniques of stabilization theory, yield Brownian sheet limits for the defect volume and mean width functionals. Finally we provide explicit variance asymptotics and central limit theorems for the k-face and intrinsic volume functionals.

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arXiv2012

Variance asymptotics for random polytopes in smooth convex bodies

Let $K \subset \R^d$ be a smooth convex set and let $¶_\la$ be a Poisson point process on $\R^d$ of intensity $\la$. The convex hull of $¶_\la \cap K$ is a random convex polytope $K_\la$. As $\la \to \infty$, we show that the variance of the number of $k$-dimensional faces of $K_\la$, when properly scaled, converges to a scalar multiple of the affine surface area of $K$. Similar asymptotics hold for the variance of the number of $k$-dimensional faces for the convex hull of a binomial process in $K$.

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arXiv2011

Nonparametric estimation of surface integrals

The estimation of surface integrals on the boundary of an unknown body is a challenge for nonparametric methods in statistics, with powerful applications to physics and image analysis, among other fields. Provided that one can determine whether random shots hit the body, Cuevas et al. [Ann. Statist. 35 (2007) 1031--1051] estimate the boundary measure (the boundary length for planar sets and the surface area for 3-dimensional objects) via the consideration of shots at a box containing the body. The statistics considered by these authors, as well as those in subsequent papers, are based on the estimation of Minkowski content and depend on a smoothing parameter which must be carefully chosen. For the same sampling scheme, we introduce a new approach which bypasses this issue, providing strongly consistent estimators of both the boundary measure and the surface integrals of scalar functions, provided one can collect the function values at the sample points. Examples arise in experiments in which the density of the body can be measured by physical properties of the impacts, or in situations where such quantities as temperature and humidity are observed by randomly distributed sensors. Our method is based on random Delaunay triangulations and involves a simple procedure for surface reconstruction from a dense cloud of points inside and outside the body. We obtain basic asymptotics of the estimator, perform simulations and discuss, via Google Earth's data, an application to the image analysis of the Aral Sea coast and its cliffs.

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arXiv2009

Laws of large numbers and nearest neighbor distances

We consider the sum of power weighted nearest neighbor distances in a sample of size n from a multivariate density f of possibly unbounded support. We give various criteria guaranteeing that this sum satisfies a law of large numbers for large n, correcting some inaccuracies in the literature on the way. Motivation comes partly from the problem of consistent estimation of certain entropies of f.

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