Weak laws of large numbers in geometric probability

By Mathew D. Penrose, Joseph E. Yukich. 2003

209 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/181dfa593c11bd54b6bd5a0ff569720d8623ab66

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Citations in the corpus

Multiscale methods for data on graphs and irregular multidimensional situations (2009, background): “For Voronoi tessellations the average number of neighbours is higher, nearer 6 in two dimensions for moderate numbers of points (see Penrose (1996) and Penrose and Yukich (2003)).”
Gaussian Limts for Random Geometric Measures (2007, background): “(3; 18; 21; 23; 27), along with a variety of other graphs on random points, such as directed k-nearest neighbour graphs, sphere of influence graphs and Delaunay graphs.”
Normal approximation for stabilizing functionals (2017, background): “For example, in the setting X = R, we expect that all of the statistics Hs and H ′ n described in [6, 25, 28, 29, 30] consist of sums of scores ξs and ξn satisfying the conditions of Theorem 2.”
Minimax and maximin space-filling designs: some properties and methods for construction (2017, unknown): “One may refer for instance to Redmond and Yukich (1996), Yukich (1998), Penrose and Yukich (2003), Wade (2007), Penrose and Yukich (2011) for extensions to power-weighted edge lengths, considering various types of convergence and different conditions on ϕ .”
The Objective Method: Probabilistic Combinatorial Optimization and Local Weak Convergence (2004, methodology): “In particular, we address progress on the minimal spanning tree problem, including a recent weak law of large numbers developed in Penrose and Yukich [ 53 ] which has close ties to the objective method.”
The local weak limit of the minimum spanning tree of the complete graph (2013, background): “h of the minimum spanning tree, then X n n(d 1)=d a! :s c: This law of large numbers for Euclidean MST’s is the jumping-o point for a massive amount of research: on more general laws of large numbers [69,67,75,61], on central limit theorems ([16,41,45,46,60,77], and on the large-nscaling of various ”
Measuring Association on Topological Spaces Using Kernels and Geometric Graphs (2020, background): “The proof is a consequence of combining different existing results from the theory of stabilizing graphs and stabilizing graph functionals (see [79] for details).”
The Radial Spanning Tree of a Poisson Point Process (2007, methodology): “This class was first introduced by [18] and it was used by Penrose and Yukich [23, 24]; it is slightly modified here to suit to our framework.”
Limit theory for point processes in manifolds (2011, methodology): “1) [5, 35, 36, 38, 39, 45] whereas when M = R , there is a relative dearth of results, although spatial data generated by a curved surface embedded in three dimensional space are arguably more natural than those involving data generated by flat surfaces.”
Gaussian Limits for Random Measures in Geometric Probability (2005, methodology): “The next result is the CLT counterpart to the main result of [26] and follows immediately from Theorem 2.”
Normal Approximation in Geometric Probability (2004, background): “If κ is bounded away from 0 and infinity and A is convex, then geometric arguments show that there is a random variable R with exponentially decaying tails such that for any x ∈ Pλ, the value of ξ(x;Pλ) is unaffected by points outside Bλ−1/dR(x) [4, 11, 13].”