TY - JOUR

T1 - Limit theory for point processes in manifolds

AU - Penrose, Mathew D.

AU - Yukich, J. E.

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84888319654&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1214/12-AAP897

U2 - 10.1214/12-AAP897

DO - 10.1214/12-AAP897

M3 - Article

SN - 1050-5164

VL - 23

SP - 2161

EP - 2211

JO - Annals of Applied Probability

JF - Annals of Applied Probability

IS - 6

ER -