Showing papers by "Mathew D. Penrose published in 2007"
TL;DR: In this article, a central limit theorem for bounded test functions on independent random marked vectors with a common density is given, where the measure is defined as a measure determined by the (suitably rescaled) set of points near the vertices of vertices.
Abstract: Given $n$ independent random marked $d$-vectors $X_i$ with a common density, define the measure $
u_n = \sum_i \xi_i $, where $\xi_i$ is a measure (not necessarily a point measure) determined by the (suitably rescaled) set of points near $X_i$. Technically, this means here that $\xi_i$ stabilizes with a suitable power-law decay of the tail of the radius of stabilization. For bounded test functions $f$ on $R^d$, we give a central limit theorem for $
u_n(f)$, and deduce weak convergence of $
u_n(\cdot)$, suitably scaled and centred, to a Gaussian field acting on bounded test functions. The general result is illustrated with applications to measures associated with germ-grain models, random and cooperative sequential adsorption, Voronoi tessellation and $k$-nearest neighbours graph.
105 citations
TL;DR: In this article, a measure νn = ∑iξi, where ξi is a measure (not necessarily a point measure) which stabilizes, is determined by the (suitably rescaled) set of points near Xi.
Abstract: Given n independent random marked d-vectors (points) Xi distributed with a common density, define the measure νn=∑iξi, where ξi is a measure (not necessarily a point measure) which stabilizes; this means that ξi is determined by the (suitably rescaled) set of points near Xi. For bounded test functions f on Rd, we give weak and strong laws of large numbers for νn(f). The general results are applied to demonstrate that an unknown set A in d-space can be consistently estimated, given data on which of the points Xi lie in A, by the corresponding union of Voronoi cells, answering a question raised by Khmaladze and Toronjadze. Further applications are given concerning the Gamma statistic for estimating the variance in nonparametric regression.
76 citations
TL;DR: In this paper, the Gamma statistic is used for estimating the variance in nonparametric regression and weak and strong laws of large numbers for large numbers are given for bounded test functions on R^d, where the general results are applied to demonstrate that an unknown set $A$ in $d$-space can be consistently estimated given data on which of the points $X_i$ lie in $A$, by the corresponding union of Voronoi cells, answering a question raised by Khmaladze and Toronjadze.
Abstract: Given $n$ independent random marked $d$-vectors (points) $X_i$ distributed with a common density, define the measure $
u_n=\sum_i\xi_i$, where $\xi_i$ is a measure (not necessarily a point measure) which stabilizes; this means that $\xi_i$ is determined by the (suitably rescaled) set of points near $X_i$. For bounded test functions $f$ on $R^d$, we give weak and strong laws of large numbers for $
u_n(f)$. The general results are applied to demonstrate that an unknown set $A$ in $d$-space can be consistently estimated, given data on which of the points $X_i$ lie in $A$, by the corresponding union of Voronoi cells, answering a question raised by Khmaladze and Toronjadze. Further applications are given concerning the Gamma statistic for estimating the variance in nonparametric regression.
63 citations
TL;DR: This work provides a rate of approximation to the normal and shows that the finite dimensional distributions of the packing measures converge to those of a mean zero generalized Gaussian field.
Abstract: 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 → ∞. 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.
36 citations
TL;DR: In this article, the authors give an explicit multivariate central limit theorem for the nearest-neighbour graph on Poisson points on a finite collection of disjoint intervals.
Abstract: Consider a measure $\mu_\lambda = \sum_x \xi_x \delta_x$ where the sum is over points $x$ of a Poisson point process of intensity $\lambda$ on a bounded region in $d$-space, and $\xi_x$ is a functional determined by the Poisson points near to $x$, i.e. satisfying an exponential stabilization condition, along with a moments condition (examples include statistics for proximity graphs, germ-grain models and random sequential deposition models). A known general result says the $\mu_\lambda$-measures (suitably scaled and centred) of disjoint sets in $R^d$ are asymptotically independent normals as $\lambda \to \infty$; here we give an $O(\lambda^{-1/(2d + \epsilon)})$ bound on the rate of convergence. We illustrate our result with an explicit multivariate central limit theorem for the nearest-neighbour graph on Poisson points on a finite collection of disjoint intervals.
1 citations
TL;DR: In this article, a general existence result for interacting particle systems with local interactions and bounded jump rates but noncompact state space at each site is given for jump events at a site that affect the state of its neighbours.
Abstract: We give a general existence result for interacting particle systems with local interactions and bounded jump rates but noncompact state space at each site. We allow for jump events at a site that affect the state of its neighbours. We give a law of large numbers and functional central limit theorem for additive set functions taken over an increasing family of subcubes of $Z^d$. We discuss application to marked spatial point processes with births, deaths and jumps of particles, in particular examples such as continuum and lattice ballistic deposition and a sequential model for random loose sphere packing.