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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 ℝ 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 ....

Moderate deviations for some point measures in geometric probability

Yu Baryshnikov;  P. Eichelsbacher;  T. Schreiber;  J. E. Yukich — 2008

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

Functionals in geometric probability are often expressed as sums of bounded functions exhibiting exponential stabilization. Methods based on cumulant techniques and exponential modifications of measures show that such functionals satisfy moderate deviation principles. This leads to moderate deviation principles and laws of the iterated logarithm for random packing models as well as for statistics associated with germ-grain models and nearest neighbor graphs.

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