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Porcu The Dagum family of isotropic correlation functions available at arXiv:0705.0456v1 [math.ST
"... A function ρ:[0, ∞) → (0,1] is a completely monotonic function if and only if ρ(‖x ‖ 2) is positive definite on R d for all d and thus it represents the correlation function of a weakly stationary and isotropic Gaussian random field. Radial positive definite functions are also of importance as they ..."
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A function ρ:[0, ∞) → (0,1] is a completely monotonic function if and only if ρ(‖x ‖ 2) is positive definite on R d for all d and thus it represents the correlation function of a weakly stationary and isotropic Gaussian random field. Radial positive definite functions are also of importance as they represent characteristic functions of spherically symmetric probability distributions. In this paper, we analyze the function ( β x ρ(β,γ)(x) = 1 − 1 + xβ)γ, x ≥ 0, β,γ> 0, called the Dagum function, and show those ranges for which this function is completely monotonic, that is, positive definite, on any ddimensional Euclidean space. Important relations arise with other families of completely monotonic and logarithmically completely monotonic functions.
Complete monotonicity for inverse powers of some combinatorially defined polynomials
, 2013
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Stieltjes and other integral representations for functions of
"... ( v2.0 released June 2011) We show that many functions containing the Lambert W function are Stieltjes functions. We extend the known properties of the set of Stieltjes functions and also prove a generalization of a conjecture of Jackson, Procacci & Sokal. In addition, we consider the relationship o ..."
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( v2.0 released June 2011) We show that many functions containing the Lambert W function are Stieltjes functions. We extend the known properties of the set of Stieltjes functions and also prove a generalization of a conjecture of Jackson, Procacci & Sokal. In addition, we consider the relationship of functions of W to the class of completely monotonic functions and show that W is a complete Bernstein function.