The Dagum family of isotropic correlation functions

The Dagum family of isotropic correlation functions

Christian Berg, Jorge Mateu, and Emilio Porcu

Source: Bernoulli Volume 14, Number 4 (2008), 1134-1149.

Abstract

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

$\rho(\beta ,\gamma)(x)=1-\biggl(\frac{x^{\beta}}{1+x^{\beta}}\biggr)^{\gamma},\qquad x\ge 0,\ \beta,\gamma>0,$

called the Dagum function, and show those ranges for which this function is completely monotonic, that is, positive definite, on any d-dimensional Euclidean space. Important relations arise with other families of completely monotonic and logarithmically completely monotonic functions.

Keywords: Bernstein function; completely monotonic function; Dagum family; isotropy; logarithmically completely monotonic function; Stieltjes transform

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.bj/1225980574
Digital Object Identifier: doi:10.3150/08-BEJ139
Zentralblatt MATH identifier: 1158.60350
Mathematical Reviews number (MathSciNet): MR2543589

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