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 -