Laws of large numbers in stochastic geometry with statistical applications

By Mathew D. Penrose. 2007

79 citations recorded by the source corpus.

Given $n$ independent random marked $d$-vectors (points) $X_i$ distributed with a common density, define the measure $\nu_n=\sum_i\xi_i$, 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 $\nu_n(f)$. 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.

Subfields: Environmental Engineering, Geometry and Topology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/1f88a34fa8eb9688e9ff1b3c63180a2a42815df4

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In Environmental Engineering

Joseph E. Yukich

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Citations in the corpus

Normal approximation for stabilizing functionals (2017, background): “∩ A) if x∈ Ac. (5.12) Deﬁne ν± s (x,X) := sν±(x,X). By the deﬁnition of ν± at (5.12) we have sVol(As) = X x∈Ps ν− s (x,Ps)+sVol(A) and sVol(A∆As) = X x∈Ps ν+ s (x,Ps). The arguments of Section 5.1 of [25] show that the scores ν± s have a radius of stabilization Rs(x,Ps ∪ {x}) with respect to Ps whic”
Gaussian Limts for Random Geometric Measures (2007, methodology): “Recently, attention has turned to the asymptotic behaviour of the measure itself (rather than only its total measure), notably in (3; 11; 18; 19; 24; 25).”
Limit theory for point processes in manifolds (2011, methodology): “2 of [36] there is a coupling in which the random variables under the expectations on the left and right-hand sides of (5.”