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Clustering percolation and directionally convex ordering of point processes

153 pages
Clustering, percolation and directionally convex ordering of point processes B. B?aszczyszyn ENS/INRIA, Paris, France blaszczy joint work with D. Yogeshwaran Summer Academy on Stochastic Analysis, Modelling and Simulation of Complex Structures Sollerhaus, Hirschegg/Kleinwalsertal, 11–24 September 2011 – p. 1

  • stochastic analysis

  • blaszczy joint

  • usual probabilistic

  • b?aszczyszyn ens

  • complex structures

  • mean measure

  • euclidean space


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Clustering, percolation and directionally
convex ordering of point processes
B. Błaszczyszyn
ENS/INRIA, Paris, France
www.di.ens.fr/ blaszczy
joint work with D. Yogeshwaran
Summer Academy on Stochastic Analysis, Modelling
and Simulation of Complex Structures
¨
Sollerhaus, Hirschegg/Kleinwalsertal, 11–24 September 2011
– p. 1Point process
Point process: random, locally finite, “pattern of points” Φ
in some spaceE.
A realization of Φ
2
onE =R .
– p. 2Point process; cont’d
Usual probabilistic formalism:
Φ is a measurable mapping from a probability space
(Ω,A, P) to a measurable spaceM “of point patterns”,
d
say, on Euclidean spaceE =R of dimensiond≥ 1.
– p. 3Point process; cont’d
Usual probabilistic formalism:
Φ is a measurable mapping from a probability space
(Ω,A, P) to a measurable spaceM “of point patterns”,
d
say, on Euclidean spaceE =R of dimensiond≥ 1.
A point pattern is considered as a counting measure; its
points are atoms of this measure. Hence
Φ(B) = (random) number of points of Φ in setB
for every measurable (Borel) subsetB⊂E.
– p. 3Point process; cont’d
Usual probabilistic formalism:
Φ is a measurable mapping from a probability space
(Ω,A, P) to a measurable spaceM “of point patterns”,
d
say, on Euclidean spaceE =R of dimensiond≥ 1.
A point pattern is considered as a counting measure; its
points are atoms of this measure. Hence
Φ(B) = (random) number of points of Φ in setB
for every measurable (Borel) subsetB⊂E.
Mean measure of Φ:
E(Φ(B)) = expected number of points of Φ inB.
– p. 3Clustering of points
Clustering in a point pattern roughly means that the points
lie in clusters (groups) with the clusters being spaced out.
– p. 4Clustering of points
Clustering in a point pattern roughly means that the points
lie in clusters (groups) with the clusters being spaced out.
How to compare clustering properties of two point
processes (pp) Φ , Φ having “on average” the same
1 2
number of points per unit of space?
– p. 4Clustering of points
Clustering in a point pattern roughly means that the points
lie in clusters (groups) with the clusters being spaced out.
How to compare clustering properties of two point
processes (pp) Φ , Φ having “on average” the same
1 2
number of points per unit of space?
More precisely, having the same mean measure:
E(Φ (B)) = E(Φ (B)) for allB⊂E.
1 2
– p. 4Stochastic comparison of point processes
But how do we compare random objects (their distributions)?
– p. 5Stochastic comparison of point processes
But how do we compare random objects (their distributions)?
→ stochastic orderings (to be explained).
– p. 5