Limit theory for point processes in manifolds

Mathew D. Penrose, J. E. Yukich

Research output: Contribution to journalArticlepeer-review

24 Citations (Scopus)

Abstract

Let Yi, i ≥ 1, be i.i.d. random variables having values in an mdimensional manifold M ⊂ Rd and consider sums ni =1 ξ(n1/mYi , {n1/mYj }nj =1), where ξ is a real valued function defined on pairs (y,Y), with y ∈ Rd and Y ⊂ Rd locally finite. Subject to ξ 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 on homogeneous Poisson point processes on m-dimensional hyperplanes tangent to 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 {Yi }n i =1.
Original languageEnglish
Pages (from-to)2161-2211
Number of pages52
JournalAnnals of Applied Probability
Volume23
Issue number6
DOIs
Publication statusPublished - 2013

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