an:05833874
Zbl 1213.60096
Bierm??, Hermine; Estrade, Anne; Kaj, Ingemar
Self-similar random fields and rescaled random balls models
EN
J. Theor. Probab. 23, No. 4, 1110-1141 (2010).
00271105
2010
j
60G60 60G55 60G18 60D05 60G20 60F05
self-similarity; generalized random field; Poisson point process; fractional Poisson field; fractional Brownian field
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.
Wolfgang Freudenberg (Cottbus)