Modelling spatio-temporal data: A new variogram and covariance structure proposal

Lamb's problem on random mass density fields with fractal and Hurst effects

“…Considering that a wide range of natural phenomena including geological formations (e.g. [3]) possess fractal and Hurst characteristics, two very interesting RF models have been developed over the past 20 years: Cauchy & Dagum [5,21]. In general, the fractal dimension can be described as a roughness measure that ranges from 2 to 3 in value for two-dimensional systems.…”

Critical markov branching process limit theorems allowing infinite variance

“…This property trivially follows from Theorem 3.2 if 0 < α ≤ 1 and δ > 0. Porcu et al (2007) showed complete monotonicity for other parameter configurations, and their results were extended in Berg et al (2008).…”

“…Let X α,δ denote a random variable having the Laplace-Stieltjes transform ψ α,δ specified in (3.5) and distribution function D α,δ . The function ψ α,δ with arbitrary positive values of α and δ has been postulated in Porcu et al (2007) as a candidate variogram for spatio-temporal modelling. For this application, it is necessary to determine for which parameter values ψ α,δ is completely monotone.…”

Continuum mechanics beyond the second law of thermodynamics

“…Special cases of this class are: C C (·, 2, γ ) which is the characteristic function of the symmetric Bessel distribution; C C (·, α, α) which is the characteristic function of the Linnik distribution and C C (·, 1, γ ) which is the symmetric Generalized Linnik characteristic function [31]. The Dagum (D) random field [29,32], also allowing independent treatment of D and H, is specified by 35) where 0 < < δ and 0 < δ ≤ 2 are sufficient conditions for positive definiteness, δ is the smoothing parameter and is the Hurst effect parameter.…”