Limit theory for point processes in manifolds

Abstract

Let $Y_i,i\ge 1$, be i.i.d. random variables having values in an $m$-dimensional manifold $\mathcal{M}\subset {\mathbb{R}}^d$ and consider sums ${\sum }_{i=1}^n\xi (n^{1/m}{Y}_i,\{n^{1/m}{Y}_j{\}}_{j=1}^n)$, where $\xi$ 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 $\xi$ 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 $\xi$ 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 $\{Y_i{\}}_{i=1}^n$.

Published in at http://dx.doi.org/10.1214/12-AAP897 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (8)

• A class of Rényi information estimators for multidimensional densities
• A new method of normal approximation
• Random geometric complexes
• Gaussian limits for random measures in geometric probability
• Laws of large numbers in stochastic geometry with statistical applications
• Limit theory for point processes in manifolds
• Gaussian limits for generalized spacings
• Explicit laws of large numbers for random nearest-neighbour type graphs

Cited by in corpus (4)

• Limit theory for point processes in manifolds
• Strong law of large numbers for Betti numbers in the thermodynamic regime
• Surface order scaling in stochastic geometry
• Nonhomogeneous Euclidean first-passage percolation and distance learning