159 research outputs found

    Variance asymptotics and scaling limits for Gaussian Polytopes

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    Let KnK_n be the convex hull of i.i.d. random variables distributed according to the standard normal distribution on Rd\R^d. We establish variance asymptotics as nn \to \infty for the re-scaled intrinsic volumes and kk-face functionals of KnK_n, k{0,1,...,d1}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 Rd1×R\R^{d-1} \times \R with intensity ehdhdve^h dh dv. The scaling limit of the boundary of KnK_n as nn \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

    Singularity points for first passage percolation

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    Let 0<a<b<0<a<b<\infty be fixed scalars. Assign independently to each edge in the lattice Z2\mathbb{Z}^2 the value aa with probability pp or the value bb with probability 1p1-p. For all u,vZ2u,v\in\mathbb{Z}^2, let T(u,v)T(u,v) denote the first passage time between uu and vv. We show that there are points xR2x\in\mathbb{R}^2 such that the ``time constant'' in the direction of xx, namely, limnn1Ep[T(0,nx)],\lim_{n\to\infty}n^{-1}\mathbf{E}_p[T(\mathbf{0},nx)], is not a three times differentiable function of pp.Comment: Published at http://dx.doi.org/10.1214/009117905000000819 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Variance Asymptotics and Scaling Limits for Random Polytopes

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    Let K be a convex set in R d and let K λ\lambda be the convex hull of a homogeneous Poisson point process P λ\lambda of intensity λ\lambda on K. When K is a simple polytope, we establish scaling limits as λ\lambda \rightarrow \infty for the boundary of K λ\lambda in a vicinity of a vertex of K and we give variance asymptotics for the volume and k-face functional of K λ\lambda, k \in {0, 1, ..., d -- 1}, resolving an open question posed in [18]. The scaling limit of the boundary of K λ\lambda 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

    Limit theory for point processes in manifolds

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    Let Yi,i1Y_i,i\geq1, be i.i.d. random variables having values in an mm-dimensional manifold MRd\mathcal {M}\subset \mathbb{R}^d and consider sums i=1nξ(n1/mYi,{n1/mYj}j=1n)\sum_{i=1}^n\xi(n^{1/m}Y_i,\{n^{1/m}Y_j\}_{j=1}^n), where ξ\xi is a real valued function defined on pairs (y,Y)(y,\mathcal {Y}), with yRdy\in \mathbb{R}^d and YRd\mathcal {Y}\subset \mathbb{R}^d locally finite. Subject to ξ\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 ξ\xi on homogeneous Poisson point processes on mm-dimensional hyperplanes tangent to M\mathcal {M}. We apply the general results to establish the limit theory of dimension and volume content estimators, R\'{e}nyi and Shannon entropy estimators and clique counts in the Vietoris-Rips complex on {Yi}i=1n\{Y_i\}_{i=1}^n.Comment: 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
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