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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 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 ....

Limit theory for some positive stationary processes with infinite mean

Jon Aaronson, Roland Zweimüller (2014)

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

We prove stable limit theorems and one-sided laws of the iterated logarithm for a class of positive, mixing, stationary, stochastic processes which contains those obtained from nonintegrable observables over certain piecewise expanding maps. This is done by extending Darling–Kac theory to a suitable family of infinite measure preserving transformations.

Limiting spectral distribution of XX' matrices

Arup Bose, Sreela Gangopadhyay, Arnab Sen (2010)

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

The methods to establish the limiting spectral distribution (LSD) of large dimensional random matrices includes the well-known moment method which invokes the trace formula. Its success has been demonstrated in several types of matrices such as the Wigner matrix and the sample covariance matrix. In a recent article Bryc, Dembo and Jiang [Ann. Probab.34 (2006) 1–38] establish the LSD for random Toeplitz and Hankel matrices using the moment method. They perform the necessary counting of terms in the...

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