Scaling limits for point random fields

Abstract

If X is a point random field on R d then convergence in distribution of the renormalization C λ | X λ − α λ | as λ → ∞ to generalized random fields is examined, where C λ > 0, α λ are real numbers for λ > 0, and X λ ( f) = λ − d X( f λ ) for f λ(x) = f( x λ ) . If such a scaling limit exists then C λ = λ θ g( λ), where g is a slowly varying function, and the scaling limit is self-similar with exponent θ. The classical case occurs when θ = d 2 and the limit process is a Gaussian white noise. Scaling limits of subordinated Poisson (doubly stochastic) point random fields are calculated in terms of the scaling limit of the environment (driving random field). If the exponent of the scaling limit is θ = d 2 then the limit is an independent sum of the scaling limit of the environment and a Gaussian white noise. If θ < d 2 the scaling limit coincides with that of the environment while if θ > d 2 the limit is Gaussian white noise. Analogous results are derived for cluster processes as well.