arXiv subjects
Publications and source records attributed to Mathew D. Penrose.
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.
We consider a Poisson process $η$ on an arbitrary measurable space with an arbitrary sigma-finite intensity measure. We establish an explicit Fock space representation of square integrable functions of $η$. As a consequence we identify explicitly, in terms of iterated difference operators, the integrands in the Wiener-Ito chaos expansion. We apply these results to extend well-known variance inequalities for homogeneous Poisson processes on the line to the general Poisson case. The Poincare inequality is a special case. Further applications are covariance identities for Poisson processes on (strictly) ordered spaces and Harris-FKG-inequalities for monotone functions of $η$.
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.
Consider throwing $n$ balls at random into $m$ urns, each ball landing in urn $i$ with probability $p_i$. Let $S$ be the resulting number of singletons, i.e., urns containing just one ball. We give an error bound for the Kolmogorov distance from $S$ to the normal, and estimates on its variance. These show that if $n$, $m$ and $(p_i, 1 \leq i \leq m)$ vary in such a way that $\sup_i p_i = O(n^{-1})$, then $S$ satisfies a CLT if and only if $n^2 \sum_i p_i^2$ tends to infinity, and demonstrate an optimal rate of convergence in the CLT in this case. In the uniform case $(p_i \equiv m^{-1}) with $m$ and $n$ growing proportionately, we provide bounds with better asymptotic constants. The proof of the error bounds are based on Stein's method via size-biased couplings.
Suppose that under the action of gravity, liquid drains through the unit $d$-cube via a minimal-length network of channels constrained to pass through random sites and to flow with nonnegative component in one of the canonical orthogonal basis directions of $\R^d$, $d \geq 2$. The resulting network is a version of the so-called minimal directed spanning tree. We give laws of large numbers and convergence in distribution results on the large-sample asymptotic behaviour of the total power-weighted edge-length of the network on uniform random points in $(0,1)^d$. The distributional results exhibit a weight-dependent phase transition between Gaussian and boundary-effect-derived distributions. These boundary contributions are characterized in terms of limits of the so-called on-line nearest-neighbour graph, a natural model of spatial network evolution, for which we also present some new results. Also, we give a convergence in distribution result for the length of the longest edge in the drainage network; when $d=2$, the limit is expressed in terms of Dickman-type variables.
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.
Given $n$ independent random marked $d$-vectors (points) $X_i$ distributed with a common density, define the measure $ν_n=\sum_iξ_i$, where $ξ_i$ is a measure (not necessarily a point measure) which stabilizes; this means that $ξ_i$ 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 $ν_n(f)$. 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.
Consider a measure $μ_λ= \sum_x ξ_x δ_x$ where the sum is over points $x$ of a Poisson point process of intensity $λ$ on a bounded region in $d$-space, and $ξ_x$ 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 $μ_λ$-measures (suitably scaled and centred) of disjoint sets in $R^d$ are asymptotically independent normals as $λ\to \infty$; here we give an $O(λ^{-1/(2d + ε)})$ 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.
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.
In ballistic deposition (BD), $(d+1)$-dimensional particles fall sequentially at random towards an initially flat, large but bounded $d$-dimensional surface, and each particle sticks to the first point of contact. For both lattice and continuum BD, a law of large numbers in the thermodynamic limit establishes convergence of the mean height and surface width of the interface to constants $h(t)$ and $w(t)$, respectively, depending on time $t$. We show that $h(t)$ is asymptotically linear in $t$, while $w(t)$ grows at least logarithmically in $t$ when $d=1$. We also give duality results saying that the height above the origin for deposition onto an initially flat surface is equidistributed with the maximum height for deposition onto a surface growing from a single site.
In the on-line nearest-neighbour graph (ONG), each point after the first in a sequence of points in R^d is joined by an edge to its nearest-neighbour amongst those points that precede it in the sequence. We study the large-sample asymptotic behaviour of the total power-weighted length of the ONG on uniform random points in (0,1)^d. In particular, for d=1 and weight exponent α>1/2, the limiting distribution of the centred total weight is characterized by a distributional fixed-point equation. As an ancillary result, we give exact expressions for the expectation and variance of the standard nearest-neighbour (directed) graph on uniform random points in the unit interval.
Given $n$ independent random marked $d$-vectors $X_i$ with a common density, define the measure $ν_n = \sum_i ξ_i $, where $ξ_i$ is a measure (not necessarily a point measure) determined by the (suitably rescaled) set of points near $X_i$. Technically, this means here that $ξ_i$ 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 law of large numbers and central limit theorem for $ν_n(f)$. The latter implies weak convergence of $ν_n(\cdot)$, suitably scaled and centred, to a Gaussian field acting on bounded test functions. The general result is illustrated with applications including the volume and surface measure of germ-grain models with unbounded grain sizes.
We consider random sequential adsorption processes where the initially empty sites of a graph are irreversibly occupied, in random order, either by monomers which block neighboring sites, or by dimers. We also consider a process where initially occupied sites annihilate their neighbors at random times. We verify that these processes are well defined on infinite graphs, and derive forward equations governing joint vacancy/occupation probabilities. Using these, we derive exact formulae for occupation probabilities and pair correlations in Bethe lattices. For the blocking and annihilation processes we also prove positive correlations between sites an even distance apart, and for blocking we derive rigorous lower bounds for the site occupation probability in lattices, including a lower bound of 1/3 for Z^2. We also give normal approximation results for the number of occupied sites in a large finite graph.
In Bhatt and Roy's minimal directed spanning tree (MDST) construction for a random partially ordered set of points in the unit square,all edges must respect the ``coordinatewise'' partial order and there must be a directed path from each vertex to a minimal element. We study the asymptotic behaviour of the total length of this graph with power weighted edges. The limiting distribution is given by the sum of a normal component away from the boundary and a contribution introduced by the boundary effects, which can be characterized by a fixed point equation, and is reminiscent of limits arising in the probabilistic analysis of certain algorithms. As the exponent of the power weighting increases, the distribution undergoes a phase transition from the normal contribution being dominant to the boundary effects dominating. In the critical case where the weight is simple Euclidean length, both effects contribute significantly to the limit law. We also give a law of large numbers for the total weight of the graph.
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.
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.
We consider a deposition model in which balls rain down at random towards a 2-dimensional surface, roll downwards over existing adsorbed balls, are adsorbed if they reach the surface, and discarded if not. We prove a spatial law of large numbers and central limit theorem for the ultimate number of balls adsorbed onto a large toroidal surface, and also for the number of balls adsorbed on the restriction to a large region of an infinite surface.
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.