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Laslett’s transform for the Boolean model in d

Rostislav Černý (2006)

Kybernetika

Consider a stationary Boolean model X with convex grains in d and let any exposed lower tangent point of X be shifted towards the hyperplane N 0 = { x d : x 1 = 0 } by the length of the part of the segment between the point and its projection onto the N 0 covered by X . The resulting point process in the halfspace (the Laslett’s transform of X ) is known to be stationary Poisson and of the same intensity as the original Boolean model. This result was first formulated for the planar Boolean model (see N. Cressie [Cressie])...

Limit theorems for geometric functionals of Gibbs point processes

T. Schreiber, J. E. Yukich (2013)

Annales de l'I.H.P. Probabilités et statistiques

Observations are made on a point process 𝛯 in d in a window Q λ of volume λ . The observation, or ‘score’ at a point x , here denoted ξ ( x , 𝛯 ) , is a function of the points within a random distance of x . When the input 𝛯 is a Poisson or binomial point process, the large λ limit theory for the total score x 𝛯 Q λ ξ ( x , 𝛯 Q λ ) , when properly scaled and centered, is well understood. In this paper we establish general laws of large numbers, variance asymptotics, and central limit theorems for the total score for Gibbsian input 𝛯 ....

Limsup random fractals.

Khoshnevisan, Davar, Peres, Yuval, Xiao, Yimin (2000)

Electronic Journal of Probability [electronic only]

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