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Self-similar random fields and rescaled random balls models. (English) Zbl 1213.60096
Author’s abstract: We study generalized random fields which arise as rescaling limits of spatial configurations of uniformly scattered random balls as the mean radius of the balls tends to 0 or infinity. Assuming that the radius distribution has a power-law behavior, we prove that the centered and renormalized random balls field admits a limit with self-similarity properties. Our main result states that all self-similar, translation- and rotation-invariant Gaussian fields can be obtained through a unified zooming procedure starting from a random balls model. This approach has to be understood as a microscopic description of macroscopic properties. Under specific assumptions, we also get a Poisson-type asymptotic field. In addition to investigating stationarity and self-similarity properties, we give \(L ^{2}\)-representations of the asymptotic generalized random fields viewed as continuous random linear functionals.

MSC:
60G60 Random fields
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60G18 Self-similar stochastic processes
60D05 Geometric probability and stochastic geometry
60G20 Generalized stochastic processes
60F05 Central limit and other weak theorems
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