Limit theory for point processes in manifolds

By Mathew D. Penrose, Joseph E. Yukich. 2011

64 citations recorded by the source corpus.

Let Yi;i ‚ 1; be i.i.d. random variables having values in an m-dimensional manifold M ‰ R d and consider sums P n=1 »(n 1=m Yi;fn 1=m Yig n=1 ), where » is a real valued function deflned on pairs (y;Y), with y 2R d and Y ‰R d locally flnite. Subject to » 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 » on homogeneous Poisson point processes on m-dimensional hyperplanes tangent to M. We apply the general results to establish the limit theory of

Subfields: Applied Mathematics, Geometry and Topology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/9e21624c98058cbc29fb952a5d4d0c134d23de51

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In Applied Mathematics

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Citations in the corpus

The topology of probability distributions on manifolds (2013, background): “In topological data analysis there has been work on understanding properties of random abstract simplicial complexes generated from stochastic processes [1, 2, 5, 13, 28, 29, 30, 32, 46, 47] and non-asymptotic bounds on the convergence or consistency of topological summaries as the number of points ”
Normal approximation for stabilizing functionals (2017, background): “When X is a sub-manifold of R equipped with the Euclidean metric on R, the paper [31] develops the limit theory for L NGk(Ps), s → ∞, and L (q) NGk (Xn), n → ∞.”
On the Kozachenko-Leonenko entropy estimator (2016, background): “The work of Bickel and Breiman has been generalized in many directions, see Chatterjee [5], Penrose and Yukich [19] and Biau and Devroye [3], with new ideas of proofs.”