arXiv subjects

J. E. Yukich

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

29 records · Page 2Linked to original sources

arXiv2009

Gaussian limits for generalized spacings

Nearest neighbor cells in $R^d,d\in\mathbb{N}$, are used to define coefficients of divergence ($ϕ$-divergences) between continuous multivariate samples. For large sample sizes, such distances are shown to be asymptotically normal with a variance depending on the underlying point density. In $d=1$, this extends classical central limit theory for sum functions of spacings. The general results yield central limit theorems for logarithmic $k$-spacings, information gain, log-likelihood ratios and the number of pairs of sample points within a fixed distance of each other.

math.PR↗

arXiv2008

Moderate deviations for some point measures in geometric probability

Functionals in geometric probability are often expressed as sums of bounded functions exhibiting exponential stabilization. Methods based on cumulant techniques and exponential modifications of measures show that such functionals satisfy moderate deviation principles. This leads to moderate deviation principles and laws of the iterated logarithm for random packing models as well as for statistics associated with germ-grain models and $k$ nearest neighbor graphs.

math.PR↗

arXiv2008

Stabilization and limit theorems for geometric functionals of Gibbs point processes

Given a Gibbs point process $¶^Ψ$ on $\R^d$ having a weak enough potential $Ψ$, we consider the random measures $μ_\la := \sum_{x \in ¶^Ψ \cap Q_\la} ξ(x, ¶^Ψ \cap Q_\la) δ_{x/\la^{1/d}}$, where $Q_{\la} := [-\la^{1/d}/2,\la^{1/d}/2]^d$ is the volume $\la$ cube and where $ξ(\cdot,\cdot)$ is a translation invariant stabilizing functional. Subject to $Ψ$ satisfying a localization property and translation invariance, we establish weak laws of large numbers for $\la^{-1} μ_\la(f)$, $f$ a bounded test function on $\R^d$, and weak convergence of $\la^{-1/2} μ_\la(f),$ suitably centered, to a Gaussian field acting on bounded test functions. The result yields limit laws for geometric functionals on Gibbs point processes including the Strauss and area interaction point processes as well as more general point processes defined by the Widom-Rowlinson and hard-core model. We provide applications to random sequential packing on Gibbsian input, to functionals of Euclidean graphs, networks, and percolation models on Gibbsian input, and to quantization via Gibbsian input.

math.PR↗

arXiv2008

Variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points

We show that the random point measures induced by vertices in the convex hull of a Poisson sample on the unit ball, when properly scaled and centered, converge to those of a mean zero Gaussian field. We establish limiting variance and covariance asymptotics in terms of the density of the Poisson sample. Similar results hold for the point measures induced by the maximal points in a Poisson sample. The approach involves introducing a generalized spatial birth growth process allowing for cell overlap.

math.PR↗

arXiv2006

Gaussian limits for multidimensional random sequential packing at saturation (extended version)

Consider the random sequential packing model with infinite input and in any dimension. When the input consists of non-zero volume convex solids we show that the total number of solids accepted over cubes of volume $λ$ is asymptotically normal as $λ\to \infty$. We provide a rate of approximation to the normal and show that the finite dimensional distributions of the packing measures converge to those of a mean zero generalized Gaussian field. The method of proof involves showing that the collection of accepted solids satisfies the weak spatial dependence condition known as stabilization.

math.PR↗

arXiv2006

Singularity points for first passage percolation

Let $0<a<b<\infty$ be fixed scalars. Assign independently to each edge in the lattice $\mathbb{Z}^2$ the value $a$ with probability $p$ or the value $b$ with probability $1-p$. For all $u,v\in\mathbb{Z}^2$, let $T(u,v)$ denote the first passage time between $u$ and $v$. We show that there are points $x\in\mathbb{R}^2$ such that the ``time constant'' in the direction of $x$, namely, $\lim_{n\to\infty}n^{-1}\mathbf{E}_p[T(\mathbf{0},nx)],$ is not a three times differentiable function of $p$.

math.PR↗

arXiv2005

Gaussian limits for random measures in geometric probability

We establish Gaussian limits for general measures induced by binomial and Poisson point processes in d-dimensional space. The limiting Gaussian field has a covariance functional which depends on the density of the point process. The general results are used to deduce central limit theorems for measures induced by random graphs (nearest neighbor, Voronoi and sphere of influence graph), random sequential packing models (ballistic deposition and spatial birth-growth models) and statistics of germ-grain models.

math.PR↗

arXiv2004

Normal Approximation in Geometric Probability

We use Stein's method to obtain bounds on the rate of convergence for a class of statistics in geometric probability obtained as a sum of contributions from Poisson points which are exponentially stabilizing, i.e. locally determined in a certain sense. Examples include statistics such as total edge length and total number of edges of graphs in computational geometry and the total number of particles accepted in random sequential packing models. These rates also apply to the 1-dimensional marginals of the random measures associated with these statistics.

math.PR↗

arXiv2002

Gaussian fields and random packing

Consider sequential packing of unit balls in a large cube, as in the Renyi car-parking model, but in any dimension and with Poisson input. We show after suitable rescaling that the spatial distribution of packed balls tends to that of a Gaussian field in the thermodynamic limit. We prove analogous results for related applied models, including ballistic deposition and spatial birth-growth models.

math.PR↗

arXiv2001

Weak Laws in Geometric Probability

Using a coupling argument, we establish a general weak law of large numbers for functionals of binomial point processes in d-dimensional space, with a limit that depends explicitly on the (possibly non-uniform) density of the point process. The general result is applied to the minimal spanning tree, the k-nearest neighbors graph, the Voronoi graph, and the sphere of influence graph. Functionals of interest include total edge length with arbitrary weighting, number of vertices of specifed degree, and number of components. We also obtain weak laws for functionals of marked point processes, including statistics of Boolean models.

math.PR↗

arXiv2001

Mathematics of random growing interfaces

We establish a thermodynamic limit and Gaussian fluctuations for the height and surface width of the random interface formed by the deposition of particles on surfaces. The results hold for the standard ballistic deposition model as well as the surface relaxation model in the off-lattice setting. The results are proved with the aid of general limit theorems for stabilizing functionals of marked Poisson point processes.

cond-mat.stat-mech↗