A Deep Look into the Dagum Family of Isotropic
Covariance Functions

Abstract

The Dagum family of isotropic covariance functions has two parameters that allow for decoupling of the fractal dimension and Hurst effect for Gaussian random fields that are stationary and isotropic over Euclidean spaces.

Sufficient conditions that allow for positive definiteness in $\mathbb{R}^{d}$ of the Dagum family have been proposed on the basis of the fact that the Dagum family allows for complete monotonicity under some parameter restrictions.

The spectral properties of the Dagum family have been inspected to a very limited extent only, and this paper gives insight into this direction. Specifically, we study finite and asymptotic properties of the isotropic spectral density (intended as the Hankel transform) of the Dagum model. Also, we establish some closed forms expressions for the Dagum spectral density in terms of the Fox–Wright functions. Finally, we provide asymptotic properties for such a class of spectral densities.

keywords:

Hankel Transforms; Mellin–Barnes Transforms; Spectral Theory; Positive Definite.

\authornames

T. FAOUZI ET AL

\authorone

[University of Bio Bio]Tarik Faouzi \authortwo[KU $\&$ Trinity College Dublin]Emilio Porcu \authorthree[University of Bio Bio]Igor Kondrashuk \authorfour[Mälardalen University]Anatoliy Malyarenko

\addressone

Department of Statistics, University of Bio Bio, Concepción, Chile \emailone[email protected] \addresstwoDepartment of Mathematics, Khalifa University at Abu Dhabi, $\&$ School of Computer Science and Statistics, Trinity College Dublin \emailtwo[email protected] \addressthree Grupo de Matemática Aplicada, Departamento de Ciencias Básicas, Universidad del Bío-Bío, Campus Fernando May, Av. Andres Bello 720, Casilla 447, Chillán, Chile. \emailthree[email protected] \addressfourDivision of Mathematics and Physics, Mälardalen University, Box 883, 721 23 Västerås, Sweden \emailfour[email protected]

\ams

60G6044A15

1 Introduction

Isotropic covariance functions have a long history that can be traced back to [14] and [26]. The mathematical machinery that allows to implement isotropic covariance functions is based on positive definite functions, that are radially symmetric over $d$ -dimensional Euclidean spaces. In particular, the characterization of the radial part of radially symmetric positive definite functions was provided in the tour de force by [20]. There is a rich catalogue of isotropic covariance functions that are obtained by composing a given parametric family of functions defined on the positive real line with the Euclidean norm in a $d$ -dimensional Euclidean space [5]. What makes them interesting is that some parameters have a corresponding interpretation in terms of geometric properties of Gaussian random fields. For instance, the Matérn family has a parameter that allows to verify the mean square differentiability of its corresponding Gaussian random field, in concert with its fractal dimension.

The Dagum parametric family of functions was originally proposed by [15] as a new family of isotropic covariance functions associated with Gaussian random fields that are weakly stationary and isotropic over $d$ -dimensional Euclidean spaces. Sufficient conditions for positive definiteness in $\mathbb{R}^{3}$ have been provided by [17] on the basis of a criterion of the Pólya type [8]. Later, [13] have shown that the Dagum family allows for decoupling fractal dimension and Hurst effect, allowing to avoid self-similar random fields, and consequently all the issues that are related to the estimation of fractal dimension and long memory parameters under self similarity [10].

Positive definiteness of a given radial function over all $d$ -dimensional Euclidean spaces is equivalent to complete monotonicity of its radial part [20]. [3] have proved sufficient conditions for complete monotonicity of the Dagum class. Some necessary conditions have also been provided therein, but unfortunately these do not match with the sufficient ones. Hence, a complete characterization for the Dagum function is, up to date, still elusive.

A wealth of applications in applied branches of science has shown how the Dagum family can be used to model temporal or spatial phenomena where local properties (fractal dimension) and global ones (Hurst effect) are decoupled, and we refer the reader to [21], with the references therein.

Positive definiteness of a radially symmetric function in a $d$ -dimensional Euclidean space, for a given dimension $d$ , is equivalent to its Hankel transform, called radial spectral density, being nonnegative and integrable [5]. The radial spectral density is often not available in closed form, with the notable exception of the Matérn model [22]. A big effort in this direction was provided by [12] with the Generalized Cauchy model, being also a decoupler of fractal dimension and Hurst effect.

Radial spectral densities are fundamental to spatial statistics. On the one hand, knowing at least local and global properties of a radial spectral density allows, by application of Tauberian theorems [22], to inspect the properties of the associated Gaussian field in terms of mean square differentiability, fractal dimension, Hurst effect, and reproducing kernel Hilbert spaces [18]. On the other hand, the radial spectral density covers a fundamental part of statistical inference for Gaussian fields under infill asymptotics [4, 22]. Finally, radial spectral densities are fundamental to inspect the so called screening effect, which in turns plays an important role in spatial prediction, and we refer the reader to [24] as well as to the recent work by [19].

An expression for the radial spectral density associated with the Dagum family has been elusive so far. A first attempt being made by [11], who showed that such a spectral density admits a series expansion that is absolutely convergent.

This paper provides some insights in this direction. After providing background material in Section 2, Section 3 provides the main results, which are classified into three parts: we start by deriving series expansions associated with the isotropic spectral density of the Dagum class. We then provide a closed form expression, in terms of the Fox–Wright functions, for such a class of isotropic spectral densities. Finally, we provide local and global asymptotic identities. Proofs are lengthy and technical: for a neater exposition, we deferred them to the Appendix. Section 4 concludes the paper with a short discussion.

2 Background Material

2.1 Positive Definite Radial Functions

We denote by $\{Z(\mathbf{s}),\mathbf{s}\in\mathbb{R}^{d}\}$ a centred Gaussian random field in $\mathbb{R}^{d}$ , with the stationary covariance function $C:\mathbb{R}^{d}\to\mathbb{R}$ . We consider the class $\Phi_{d}$ of continuous mappings $\phi:[0,\infty)\to\mathbb{R}$ with $\phi(0)=1$ , such that

{\rm cov}\left(Z(\mathbf{s}),Z(\mathbf{s}^{\prime})\right)=C(\mathbf{s}^{\prime}-\mathbf{s})=\phi(\|\mathbf{s}^{\prime}-\mathbf{s}\|),

with $\mathbf{s},\mathbf{s}^{\prime}\in\mathbb{R}^{d}$ , and $\|\cdot\|$ denoting the Euclidean norm. Gaussian fields with such covariance functions are called weakly stationary and isotropic. The function $C$ is called isotropic or radially symmetric, and the function $\phi$ its radial part.

[20] characterized the class $\Phi_{d}$ as being scale mixtures of the characteristic functions of random vectors uniformly distributed on the spherical shell of $\mathbb{R}^{d}$ :

\phi(r)=\int_{0}^{\infty}\Omega_{d}(r\xi)F(\mathrm{d}\xi),\qquad r\geq 0,

with $\Omega_{d}(r)=r^{-(d-2)/2}J_{(d-2)/2}(r)$ and $J_{\nu}$ a Bessel function of order $\nu$ . Here, $F$ is a probability measure. The function $\phi$ is the uniquely determined characteristic function of a random vector, $\mathbf{X}$ , such that $\mathbf{X}=\mathbf{U}\cdot R$ , where equality is intended in the distributional sense, where $\mathbf{U}$ is uniformly distributed over the spherical shell of $\mathbb{R}^{d}$ , $R$ is a nonnegative random variable with probability distribution, $F$ , and where $\mathbf{U}$ and $R$ are independent.

[5] describes the properties of the measures, $F$ , termed the Schoenberg measures there, and shows the existence of projection operators that map the elements of $\Phi_{d}$ onto the elements of $\Phi_{d^{\prime}}$ , for $d^{\prime}\neq d$ . Throughout, we adopt their illustrative name and will call the function $F$ associated with $\phi$ a Schoenberg measure. The derivative of $F$ is called the isotropic spectral density. If $\phi$ is absolutely integrable, then the Fourier inversion (the Hankel transform) becomes possible. The Fourier transforms of radial versions of the members of $\Phi_{d}$ , for a given $d$ , have a simple expression, as reported in [23] and [27]. For a member $\phi$ of the family $\Phi_{d}$ , we define its isotropic spectral density as

\widehat{\phi}(z)=\frac{z^{1-d/2}}{(2\pi)^{d/2}}\int_{0}^{\infty}u^{d/2}J_{d/2-1}(uz)\phi(u)\,{\rm d}u,\qquad z\geq 0.

The classes $\Phi_{d}$ are nested, with the inclusion relation $\Phi_{1}\supset\Phi_{2}\supset\ldots\supset\Phi_{\infty}$ being strict, and where $\Phi_{\infty}:=\bigcap_{d\geq 1}\Phi_{d}$ is the class of mappings $\phi$ whose radial version is positive definite on all $d$ -dimensional Euclidean spaces.

2.2 Parametric Families of Isotropic Covariance Functions

The Generalized Cauchy family of members of $\Phi_{\infty}$ [9] is defined as:

{\cal C}_{\delta,\lambda}(r)=\left(1+r^{\delta}\right)^{-\lambda/\delta},\qquad r\geq 0,

(1)

where the conditions $\delta\in(0,2]$ and $\lambda>0$ are necessary and sufficient for ${\cal C}_{\delta,\lambda}$ to belong to the class $\Phi_{\infty}$ . The parameter $\delta$ is crucial for the differentiability at the origin and, as a consequence, for the degree of differentiability of the associated sample paths. Specifically, for $\delta=2$ , they are infinitely differentiable and they are not differentiable for $\delta\in(0,2)$ .

For a Gaussian random field in $\mathbb{R}^{d}$ with isotropic covariance function ${\cal C}_{\delta,\lambda}(\|\cdot\|)$ , the sample paths have fractal dimension $D=d+1-\delta/2$ for $\delta\in(0,2)$ and, if $\lambda\in(0,d]$ , the long memory parameter or Hurst coefficient is identically equal to $H=1-\lambda/2$ . Thus, $D$ and $H$ may vary independently of each other [9, 12]. [6] and [12] have shown that the isotropic spectral density, $\widehat{{\cal C}}_{\delta,\lambda}$ of the Generalized Cauchy covariance function is identically equal to

\widehat{{\cal C}}_{\delta,\lambda}(z)=-\frac{z^{-d}}{2^{d/2-1}\pi^{d/2+1}}\mathrm{Im}\int_{0}^{\infty}\frac{{\cal K}_{(d-2)/2}(t)}{(1+\exp(i\frac{\pi\delta}{2})(t/z)^{\delta})^{\lambda/\delta}}t^{d/2}\,\mathrm{d}t,\qquad z\geq 0,

for $\lambda>0$ and $\delta\in(0,2)$ .

The Dagum family ${\cal D}_{\lambda,\delta}:[0,\infty)\to\mathbb{R}$ is defined as

{\cal D}_{\lambda,\delta}(r)=1-\left(\frac{r^{\delta}}{1+r^{\delta}}\right)^{\lambda},\qquad r\geq 0.

(2)

[15] and subsequently [17] show that ${\cal D}_{\lambda,\delta}$ belongs to the class $\Phi_{3}$ provided $\delta<(7-\lambda)/(1+5\lambda)$ and $\lambda<7$ . [3] have shown that ${\cal D}_{\lambda,\delta}\in\Phi_{\infty}$ if and only if the function ${\cal A}_{\delta,\lambda}$ , defined as

{\cal A}_{\delta,\lambda}(r)=\frac{r^{\delta\lambda-1}}{\left(1+r^{\delta}\right)^{\lambda+1}},\qquad r\geq 0,

belongs to $\Phi_{\infty}$ . In particular, sufficient conditions for ${\cal D}_{\lambda,\delta}\in\Phi_{\infty}$ become $\delta\lambda\leq 1$ and $\beta\leq 1$ . Also, for $\delta=1/\lambda$ we have ${\cal D}_{\lambda,1/\lambda}\in\Phi_{\infty}$ if and only if $\delta\leq 1$ .

To simplify notation, throughout we shall write ${\cal D}_{\lambda,\delta}(\boldsymbol{r})$ for ${\cal D}_{\lambda,\delta}\circ\|\boldsymbol{r}\|$ , $\boldsymbol{r}\in\mathbb{R}^{d}$ , with $\circ$ denoting composition. Similar notation will be used for ${\cal C}_{\delta,\lambda}(\boldsymbol{r})$ , $\boldsymbol{r}\in\mathbb{R}^{d}$ . Analogously, we use $\widehat{{\cal D}}_{\delta,\lambda}(\boldsymbol{z})$ for $\widehat{{\cal D}}_{\delta,\lambda}(\|\boldsymbol{z}\|)$ , $\boldsymbol{z}\in\mathbb{R}^{d}$ , and sometimes we shall make use of the notation $z$ for $\|\boldsymbol{z}\|$ . Similar notation will be used for the isotropic spectral density $\widehat{C}_{\delta,\lambda}$ .

2.3 Fractal Dimensions and the Hurst effect

The local properties of a time series or a surface of $\mathbb{R}^{d}$ are identified through the so-called fractal dimension, $D$ , which is a roughness measure with range $[d,d+1)$ , and with higher values indicating rougher surfaces. The long memory in time series or spatial data is associated with power law correlations, and often referred to as the Hurst effect. Long memory dependence is characterized by the $H$ parameter. Local and global properties of a Gaussian random field have an intimate connection with its associated isotropic covariance function. In particular, if, for some $\alpha\in(0,2]$ , the radial part $\varphi\in\Phi_{d}$ satisfies

\lim_{r\to 0}\frac{\varphi(r)}{r^{\alpha}}=1,

(3)

then the realizations of the Gaussian random field have fractal dimension $D=d+1-\alpha/2$ , with probability $1$ . Thus, estimation of $\alpha$ is linked with that of the fractal dimension $D$ . Conversely, if for some $\beta\in(0,1)$ ,

\lim_{r\to\infty}\varphi(r)r^{\beta}=1,

(4)

then the Gaussian random field is said to have long memory, with Hurst coefficient $H=1-\beta/2$ . For $H\in(1/2,1)$ or $H\in(0,1/2)$ the correlation is said to be respectively persistent or anti-persistent. In general, $D$ and $H$ are independent of each other, but under the assumption of self-affinity they find an intimate connection in the well-known linear relationship $D+H=d+1$ . The Cauchy model behaves like (3) for $\alpha=\delta\in(0,2]$ and like (4) for $\beta=\lambda\in(0,1)$ . For the reparameterized version ${\cal D}_{\lambda,\delta/\lambda}$ , we have exactly the same result. For both models the local and global behaviour parameters may be estimated independently.

For $\delta\in(0,2)$ , the Dagum covariance function can be rewritten as

{\cal D}_{\lambda,\delta}(\boldsymbol{r})=1-(1+1/\|\boldsymbol{r}\|^{\delta})^{-\lambda},\qquad\boldsymbol{r}\in\mathbb{R}^{d}.

When $\|\boldsymbol{r}\|$ is large, the Dagum covariance function has the following asymptotic behavior

{\cal D}_{\lambda,\delta}(\boldsymbol{r})\sim\lambda\|\boldsymbol{r}\|^{-\delta},\quad\text{for}\quad\delta\in(0,1).

Hence, under these parameter restrictions, a Gaussian random field has long memory with Hurst coefficient $H=1-\delta/2$ with $\delta\in(0,1)$ .

A notable fact is the following:

\int_{[0,\infty)^{d}}{\cal D}_{\lambda,\delta}(\boldsymbol{r})\text{d}^{d}\boldsymbol{r}=\frac{\pi^{d/2}}{2^{d-1}\Gamma(d/2)}\int_{0}^{\infty}r^{d-1}\left(1-\frac{1}{(1/r^{\delta}+1)^{\lambda}}\right)\text{d}r.

(5)

Furthermore,

r^{d-1}{\cal D}_{\lambda,\delta}(\boldsymbol{r})\sim r^{d-\delta-1},\qquad r\to\infty,\quad\text{with}\quad\|\boldsymbol{r}\|=r.

The above implies that the integral (5) is finite if $\delta>d$ . An alternative way to see this is to notice that Equation (24) of this paper proves that

	$\displaystyle\int_{\mathbb{R}^{d}}{\cal D}_{\lambda,\delta}(r)\text{d}^{d}\boldsymbol{r}$	$\displaystyle=\frac{\pi^{d/2}}{\Gamma(1+d/2)}\int_{0}^{\infty}r^{d-1}\left(1-\frac{1}{(1/r^{\delta}+1)^{\lambda}}\right)\text{d}r$
		$\displaystyle=\frac{\pi^{d/2}}{\Gamma(1+d/2)}\frac{{\rm B}(d/\delta+\lambda,-d/\delta)}{\delta}.$

3 Theoretical Results

3.1 Isotropic Spectral Density of the Dagum Covariance Function

This subsection aims to compute the isotropic spectral density associated with the Dagum class in $\mathbb{R}^{d}$ , for a given positive integer $d$ , when $\delta>d$ . The spectral density of Dagum class can be written as

\widehat{\cal D}_{\delta,\lambda}(\boldsymbol{z})=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}e^{-i\boldsymbol{z}\cdot\boldsymbol{r}}{\cal D}_{\lambda,\delta}(\boldsymbol{r})\,\mathrm{d}\boldsymbol{r},

with $i$ denoting the imaginary unit. [11] showed that when $\lambda=k\in 2\mathbb{Z}_{+}$ is an integer, the Dagum spectral density can be written as

\widehat{\cal D}_{\delta,k}(\boldsymbol{z})=-\displaystyle\sum_{h=0}^{k-1}(-1)^{k-h}\text{C}_{h}^{k}\widehat{\cal C}_{\delta,k-h}(\boldsymbol{z}),\qquad\boldsymbol{z}\in\mathbb{R}^{d},

where $\widehat{\cal C}_{\delta,k-h}$ is a generalized Cauchy isotropic spectral density associated with the generalized Cauchy function, ${\cal C}_{\delta,k-h}$ as defined at (1).

We start by extending this result for any $\lambda>0$ and for $d=1$ .

Theorem 3.1

For $d=1$ and $\delta>1$ , the Dagum isotropic spectral density $\widehat{D}_{\delta,\lambda}$ has the following explicit form:

\widehat{\cal D}_{\delta,\lambda}(z)=-\frac{1}{\pi}\mathrm{Im}\int_{0}^{\infty}\left[1-\frac{e^{i\delta\lambda\pi/2}r^{\delta\lambda}}{(1+e^{i\pi/2}r^{\delta})^{\lambda}}\right]e^{-zr}\,\mathrm{d}r.

To extend this result to $\mathbb{R}^{d}$ , for $d>1$ , we start by considering the case $\delta\leq d$ , $\delta\lambda\in(0,2)$ and $d>1$ .
We can show the following.

Theorem 3.2

For $d>1$ , $\delta\in(0,2)$ , and $\delta\lambda\in(0,2]$ , the Dagum isotropic spectral density $\widehat{D}_{\delta,\lambda}$ is given by

\widehat{\cal D}_{\delta,\lambda}(\boldsymbol{z})=-\frac{z^{1-\frac{d}{2}}}{2^{\frac{d}{2}-1}\pi^{\frac{d}{2}+1}}\mathrm{Im}\int_{0}^{\infty}K_{\frac{d}{2}-1}(zt)\left(1-\frac{t^{\delta\lambda}e^{i\pi\delta\lambda/2}}{(1+e^{i\pi\delta/2}t^{\delta})^{\lambda}}\right)t^{d/2}\,\mathrm{d}t,

(6)

with $z=\|\boldsymbol{z}\|.$

Albeit Theorems 3.1 and 3.2 provide some insight into understanding the isotropic spectral density associated with the Dagum covariance function, it might be desirable to have an explicit expression for such spectral densities. The following subsection attacks this problem.

3.2 Dagum spectral density expressed as Fox–Wright function

We start by defining the Fox–Wright function [7, 25] ${}_{p}\Psi_{q}$ , through the identity

_{p}\Psi_{q}\Big{[}\begin{matrix}(a_{1},A_{1})&,\cdots,&(a_{p},A_{p})\\ (b_{1},B_{1})&,\cdots,&(b_{q},B_{q})\end{matrix};-z\Big{]}=\sum_{k=0}^{\infty}\frac{(-1)^{k}\Gamma(a_{1}+kA_{1})\cdots\Gamma(a_{p}+kA_{p})}{k!\Gamma(b_{1}+kB_{1})\cdots\Gamma(b_{q}+kB_{q})}z^{k}.

(7)

It turns out that this class of special functions is intimately related to the Dagum spectral density. We formally state this fact below.

Theorem 3.3

Let $d$ be a positive integer, $\boldsymbol{z}\in\mathbb{R}^{d}$ and $z=\|\boldsymbol{z}\|$ . Let ${\cal D}_{\lambda,\delta}$ be the Dagum class as defined at (2). For $\delta\in(0,2)$ and $\delta\lambda\in(0,2)$ , the isotropic spectral density ${\cal D}_{\lambda,\delta}$ in $\mathbb{R}^{d}$ , is given by

	$\displaystyle\widehat{\cal D}_{\delta,\lambda}(\boldsymbol{z})$	$\displaystyle=\delta^{(d)}(\boldsymbol{z})-\frac{\pi^{-d/2}z^{-d}}{\Gamma(\lambda)}\,_{2}\Psi_{1}\Big{[}\begin{matrix}(\lambda,1)&(d/2,-\delta/2)\\ &(0,\delta/2)\end{matrix};-\left(\frac{z}{2}\right)^{\delta}\Big{]}$		(8)
		$\displaystyle-\frac{1}{2^{d}\pi^{d/2}}\frac{1}{\Gamma(\lambda)}\frac{2}{\delta}\,_{2}\Psi_{1}\Big{[}\begin{matrix}(\lambda+d/\delta,2/\delta)&(-d/\delta,-2/\delta)\\ &(d/2,1)\end{matrix};-\left(\frac{z}{2}\right)^{2}\Big{]}.$		(8)

where $\delta^{(d)}(\boldsymbol{z})=(2\pi)^{-d}\underset{\epsilon\to 0}{\lim}\prod_{k=1}^{d}(\frac{1}{\epsilon}\xi_{\epsilon}(z_{k}))$ , with $\xi_{\epsilon}$ being the unit impulse function, and ${}_{2}\Psi_{1}$ is the Fox–Wright function given by equation (7).

3.3 Asymptotic Properties of Dagum Spectral Density

We finish with some theoretical resuls relating to the asymptotic behaviour of the Dagum isotropic spectral density.

Theorem 3.4

For all $\delta\in(0,2)$ and $\delta\lambda\in(0,2]$ , the low frequency limit of the spectral density $\widehat{\cal D}_{\delta,\lambda}$ is given by

1.

$\widehat{\cal D}_{\delta,\lambda}(\boldsymbol{z})\sim\frac{2^{-\delta}\lambda\Gamma(d/2-\delta/2)}{\pi^{\frac{d}{2}}\Gamma(\delta/2)}z^{\delta-d}$ if $\delta\in(\frac{d-2}{2},d)$ , $z\to 0$ ;
2.

$\widehat{\cal D}_{\delta,\lambda}(\boldsymbol{z})\sim\frac{1}{\delta\pi^{\frac{d}{2}}2^{d-1}\Gamma(d/2)}\frac{\Gamma(-\mathrm{d}/\delta)\Gamma(d/\delta+\lambda)}{\Gamma(\lambda)}$ if $\delta\in(d,2)$ , $z\to 0$ .

Theorem 3.5

For $0<\delta<2$ , and $0<\delta\lambda\leq 2$ when $z\to\infty$ ,

		$\displaystyle\widehat{\cal D}_{\delta,\lambda}(\boldsymbol{z})$		(9)
		$\displaystyle\quad\sim\frac{2^{\delta\lambda}z^{-d-\delta\lambda}}{\pi^{\frac{d}{2}}\Gamma(\lambda)}\,_{3}\Psi_{2}\Big{[}\begin{matrix}(\lambda,1)&(\delta\lambda/2+1,\delta/2)&((\delta\lambda+d)/2,\delta/2)\\ &(\delta\lambda/2,\delta/2)&(1-\delta\lambda/2,-\delta/2)\end{matrix};-\left(\frac{2}{z}\right)^{\delta}\Big{]}$
		$\displaystyle\quad\sim\frac{2^{\delta\lambda-1}\lambda\delta z^{-d-\delta\lambda}}{\pi^{\frac{d}{2}}}\frac{\Gamma(d/2+\lambda\delta/2)}{\Gamma(1-\lambda\delta/2)}.$

4 Conclusion

We have obtained the expressions for the isotropic spectral density related to the Dagum family. Our results can now be used in research related to (a) best optimal unbiased linear prediction (kriging) under infill asymptotics when using the Dagum family. This in turn relies on equivalence of Gaussian measures and on the ratio between the correct and the misspecified spectral density [4]. While the Matérn covariance function has been already studied under this setting [28], the characterization of equivalence of Gaussian measures under the Dagum family has been elusive so far. Also, (b) knowing the form of the spectral density will be crucial to obtain the space-time spectral densities associated with covariance functions having a dynamical support depending on a Dagum radius, as detailed by [16].

Acknowledgements

Partial support was provided by FONDECYT grant 1130647, Chile for Emilio Porcu and by Millennium Science Initiative of the Ministry of Economy, Development, and Tourism, grant ”Millenium Nucleus Center for the Discovery of Structures in Complex Data” for Emilio Porcu. Partial support was provided by by FONDECYT grant 11200749, Chile for Tarik Faouzi and by grant Diubb 2020525 IF/R from the university of Bio Bio for Tarik Faouzi. Igor Kondrashuk was supported in part by Fondecyt (Chile) Grants Nos. 1121030, by DIUBB (Chile) Grants Nos. GI 172409/C and 181409 3/R.

Appendix A Proofs

Proof A.1 (Proof of Theorem 3.1)

\widehat{\cal D}_{\delta,\lambda}(z)=\frac{1}{\pi}\mathrm{Re}\int_{0}^{\infty}e^{i|z|t}\left(1-\frac{t^{\delta\lambda}}{(1+t^{\delta})^{\lambda}}\right)\,\mathrm{d}t.

Let $f(t;z)=e^{i|z|t}\left(1-\frac{t^{\delta\lambda}}{(1+t^{\delta})^{\lambda}}\right)$ , and consider $D_{\xi}=\{t\in\mathbb{C};|t|\leq\xi,\mathrm{Re}(t)>0,\mathrm{Im}(t)>0\}$ . Then, for $\delta\in(0,2)$ and $\delta\lambda\in(0,2)$ , $f$ is an analytic function on $D_{\xi}$ . By the Cauchy integral formula,

\oint\limits_{\partial D_{\xi}}f(t;z)\,\mathrm{d}t=0,

where $\partial D_{\xi}$ is a boundary of $D_{\xi}$ , the union of three components: the line segment $L_{1}$ along the real axis from 0 to $\xi$ , the arc $C_{\xi}$ of the circle $|t|=\xi$ from $\xi$ to $i\xi$ , and the line segment $L_{2}$ along the imaginary axis from $i\xi$ to $0$ .

Next, for any $t$ on the arc $C_{\xi}=\{t\in\mathbb{C};|t|=\xi\}$ , there is a phase $\varphi\in[0,\pi/2]$ such that $t=\xi\leavevmode\nobreak\ e^{i\varphi}$ . Then

	$\displaystyle f(t;z)$	$\displaystyle=f(\xi\leavevmode\nobreak\ e^{i\varphi};z)$
		$\displaystyle=e^{i\|z\|\xi\leavevmode\nobreak\ e^{i\varphi}}\left(1-\Big{[}\frac{\xi^{\delta}e^{i\delta\varphi}}{1+\xi^{\delta}e^{i\delta\varphi}}\Big{]}^{\lambda}\right).$

The last term of the above equality can be expressed as

1-\Big{[}\frac{\xi^{\delta}e^{i\delta\varphi}}{(1+\xi^{\delta}e^{i\delta\varphi})}\Big{]}^{\lambda}=1-\sum_{j=0}^{\infty}(-1)^{j}\frac{\Gamma(\lambda+j)}{\Gamma(\lambda)}(\xi^{-\delta}e^{-i\delta\varphi})^{j},

then

\oint\limits_{\partial D_{\xi}}f(t)\,\mathrm{d}t=-\sum_{j=1}^{\infty}(-1)^{j}\frac{\Gamma(\lambda+j)}{\Gamma(\lambda)}\xi^{-j\delta}\int_{0}^{\pi/2}e^{-i\delta\varphi j}e^{i|z|\xi e^{i\varphi}}\,\mathrm{d}\varphi.

(10)

The integral presented in Equation (10) is expressed as

	$\displaystyle\int_{0}^{\pi/2}e^{-i\delta\varphi j}e^{i\|z\|\xi e^{i\varphi}}\,\mathrm{d}\varphi=$	$\displaystyle\frac{1}{i}\int_{0}^{\pi/2}e^{-i(j\delta+1)\varphi}e^{i\|z\|\xi e^{i\varphi}}\,\mathrm{d}e^{i\varphi}$
	$\displaystyle=$	$\displaystyle\frac{1}{i}\int_{C_{1}}e^{i\|z\|\xi\omega}\omega^{-(j\delta+1)}\,\mathrm{d}\omega=\frac{(\|z\|\xi)^{j\delta}}{i}\int_{C_{\|z\|\xi}}e^{iu}u^{-(j\delta+1)}\,\mathrm{d}u,$

where $C_{1}$ is an arc of the circle $|\omega|=1$ from $1$ to $i$ and $C_{|z|\xi}$ is an arc of the circle $|u|=|z|\xi$ from $|z|\xi$ to $i|z|\xi$ .

Then,

	$\displaystyle\underset{\xi\to\infty}{\lim}\oint\limits_{\partial D_{\xi}}f(t;z)\,\mathrm{d}t=\underset{\xi\to\infty}{\lim}\int\limits_{L_{1}}f(t;x)\,\mathrm{d}t+\underset{\xi\to\infty}{\lim}\int\limits_{L_{2}}f(t;z)\,\mathrm{d}t$
	$\displaystyle+i\sum_{j=1}^{\infty}(-1)^{j}\frac{\Gamma(\lambda+j)}{\Gamma(\lambda)}\|z\|^{j\delta}\underset{\xi\to\infty}{\lim}\int_{C_{\|z\|\xi}}e^{iu}u^{-(j\delta+1)}\,\mathrm{d}u=0.$

However, we may state that

\underset{\xi\to\infty}{\lim}\int_{C_{|z|\xi}}e^{iu}u^{-(j\delta+1)}\,\mathrm{d}u=0.

(11)

Finally,

\underset{\xi\to\infty}{\lim}\int\limits_{L_{1}}f(t;x)\,\mathrm{d}t=-\underset{\xi\to\infty}{\lim}\int\limits_{L_{2}}f(t;z)\,\mathrm{d}t.

The last result implies

	$\displaystyle\widehat{\cal D}_{\delta,\lambda}(z)$	$\displaystyle=\frac{1}{\pi}\mathrm{Re}\int_{0}^{\infty}\left[1-\frac{r^{\delta\lambda}}{(1+r^{\delta})^{\lambda}}\right]e^{i\|z\|r}\,\mathrm{d}r$
		$\displaystyle=-\frac{1}{\pi}\mathrm{Im}\int_{0}^{\infty}\left[1-\frac{e^{i\delta\lambda\pi/2}r^{\delta\lambda}}{(1+e^{i\pi/2}r^{\delta})^{\lambda}}\right]e^{-\|z\|r}\,\mathrm{d}r.$

Proof A.2 (Proof of Theorem 3.2)

With $\|\boldsymbol{z}\|=z$ , we write,

		$\displaystyle\int_{\mathbb{R}^{d}}e^{i\boldsymbol{r}\boldsymbol{z}}\left(-\frac{z^{1-\frac{d}{2}}}{2^{\frac{d}{2}-1}\pi^{\frac{d}{2}+1}}\mathrm{Im}\int_{0}^{\infty}K_{\frac{d}{2}-1}(zt)\left(1-\frac{t^{\delta\lambda}e^{i\pi\delta\lambda/2}}{(1+e^{i\pi\delta/2}t^{\delta})^{\lambda}}\right)t^{d/2}\,\mathrm{d}t\right)\,\mathrm{d}\boldsymbol{z}$
		$\displaystyle=(2\pi)^{\frac{d}{2}}r^{1-\frac{d}{2}}\int_{0}^{\infty}J_{\frac{d}{2}-1}(rz)z^{\frac{d}{2}}\left(-\frac{z^{1-\frac{d}{2}}}{2^{\frac{d}{2}-1}\pi^{\frac{d}{2}+1}}\mathrm{Im}\int_{0}^{\infty}K_{\frac{d}{2}-1}(zt)\right.$
		$\displaystyle\quad\times\left.\left(1-\frac{t^{\delta\lambda}e^{i\pi\delta\lambda/2}}{(1+e^{i\pi\delta/2}t^{\delta})^{\lambda}}\right)t^{d/2}\,\mathrm{d}t\right)\,\mathrm{d}z$
		$\displaystyle=-\frac{2r^{1-\frac{d}{2}}}{\pi}\mathrm{Im}\int_{0}^{\infty}\left(1-\frac{e^{i\pi\delta\lambda/2}t^{\delta\lambda}}{(1+e^{i\pi\delta/2}t^{\delta})^{\lambda}}\right)t^{d/2}\int_{0}^{\infty}zK_{\frac{d}{2}-1}(zt)J_{\frac{d}{2}-1}(zr)\leavevmode\nobreak\ \mathrm{d}z$
		$\displaystyle=-\frac{1}{i\pi}\int_{-\infty}^{\infty}\left(1-\frac{t^{\delta\lambda}e^{i\pi\delta\lambda/2}}{(1+e^{i\pi\delta/2}t^{\delta})^{\lambda}}\right)\frac{t}{(r^{2}+t^{2})}\mathrm{d}t.$

When $\delta\in(0,2)$ and $\delta\lambda\in(0,2)$ , the last integral is expressed as

-\frac{1}{i\pi}\int_{-\infty}^{\infty}\left(1-\frac{t^{\delta\lambda}e^{i\pi\delta\lambda/2}}{(1+e^{i\pi\delta/2}t^{\delta})^{\lambda}}\right)\frac{t}{(r^{2}+t^{2})}\,\mathrm{d}t=1-\frac{r^{\delta\lambda}}{(1+r^{\delta})^{\lambda}}.

Proof A.3 (Proof of Theorem 3.3)

To find the explicit form of the Dagum spectral density, we use the Mellin–Barnes transform [2] defined through the identity

\displaystyle\frac{1}{(1+x)^{\delta}}=\frac{1}{2i\pi}\frac{1}{\Gamma(\delta)}\oint_{\Lambda}\leavevmode\nobreak\ x^{u}\Gamma(-u)\Gamma(\delta+u)\,\text{d}u,

(12)

here $\Gamma(\cdot)$ denotes the Gamma function. This representation is valid for any $x\in\mathbb{R}$ . The contour $\Lambda$ contains the vertical line which passes between left and right poles in the complex plane $u$ from negative to positive imaginary infinity, and should be closed to the left in case $x>1$ , and to the right complex infinity if $0<x<1.$

Applying Equation (12), we obtain

	$\displaystyle\widehat{{\cal D}}_{\delta,\lambda}(\boldsymbol{z})=$	$\displaystyle\frac{1}{(2\pi)^{d}}\left[\int_{\mathbb{R}^{d}}{\rm e}^{-i\boldsymbol{r}\boldsymbol{z}}\text{d}\boldsymbol{r}-\frac{1}{2i\pi}\frac{1}{\Gamma(\lambda)}\int_{\mathbb{R}^{d}}{\rm e}^{-i\boldsymbol{r}\boldsymbol{z}}\oint_{\Lambda}\Gamma(-u)\Gamma(u+\lambda)r^{-u\delta}\text{d}u\leavevmode\nobreak\ \text{d}\boldsymbol{r}\right]$
	$\displaystyle=$	$\displaystyle\frac{1}{(2\pi)^{d}}\left[\int_{\mathbb{R}^{d}}{\rm e}^{-i\boldsymbol{r}\boldsymbol{z}}\text{d}\boldsymbol{r}-\frac{1}{2i\pi}\frac{1}{\Gamma(\lambda)}\oint_{\Lambda}\Gamma(-u)\Gamma(u+\lambda)\int_{\mathbb{R}^{d}}{\rm e}^{-i\boldsymbol{r}\boldsymbol{z}}r^{-u\delta}\text{d}\boldsymbol{r}\leavevmode\nobreak\ \text{d}u\right]$

We now invoke the well known relationship [1],

\displaystyle\int_{\mathbb{R}^{d}}{\rm e}^{i\boldsymbol{x}\boldsymbol{z}}\|\boldsymbol{r}\|^{-u\delta}\text{d}\boldsymbol{r}=\frac{2^{d-u\delta}\pi^{d/2}\Gamma(d/2-u\delta/2)}{\Gamma(u\delta/2)\|\boldsymbol{z}\|^{d-u\delta}}.

Hence,

	$\displaystyle\widehat{\cal D}_{\delta,\lambda}(\boldsymbol{z})$	$\displaystyle=\delta^{(d)}(\boldsymbol{z})-\frac{1}{(2\pi)^{d}}\frac{1}{\Gamma(\lambda)}\frac{1}{2\pi i}\oint_{\Lambda}\leavevmode\nobreak\ \frac{2^{d-u\delta}\pi^{d/2}\Gamma(d/2-u\delta/2)\Gamma(-u)\Gamma\left(u+\lambda\right)}{\Gamma(u\delta/2)}\frac{1}{{z}^{d-u\delta}}\text{d}u$		(13)
		$\displaystyle=\delta^{(d)}(\boldsymbol{z})-\frac{z^{-d}}{\pi^{d/2}}\frac{1}{2\pi i}\frac{1}{\Gamma(\lambda)}\oint_{\Lambda}\frac{\Gamma(-u)\Gamma(u+\lambda)\Gamma(d/2-u\delta/2)}{\Gamma(u\delta/2)}\left(\frac{z}{2}\right)^{u\delta}\text{d}u.$		(13)

For any given value of $|z/2|,$ it is not relevant whether it is smaller or greater than $1$ . In fact, the contour might be closed to the right complex infinity. The above series is convergent for any values of the variable $z$ . The functions $u\mapsto\Gamma(-u)$ and $u\mapsto\Gamma\left(d/2-u\delta/2\right)$ contain poles in the complex plane, respectively when $-u=-n$ , and when $d/2-u\delta/2=-n,$ $n\in\mathbb{Z}_{+}$ . Using this fact and through direct inspection we obtain that the right hand side in (13) matches with (14).

	$\displaystyle\widehat{\cal D}_{\delta,\lambda}(\boldsymbol{z})$	$\displaystyle=\delta^{(d)}(\boldsymbol{z})-\frac{z^{-d}}{\pi^{d/2}}\frac{1}{\Gamma(\lambda)}\sum_{n=0}^{\infty}\frac{(-1)^{n}\ \Gamma(\lambda+n)\Gamma(d/2-n\delta/2)}{n!\Gamma(n\delta/2)}\left(\frac{z}{2}\right)^{n\delta}$		(14)
		$\displaystyle-\frac{1}{2^{d}\pi^{d/2}}\frac{1}{\Gamma(\lambda)}\frac{2}{\delta}\sum_{n=0}^{\infty}\frac{(-1)^{n}\Gamma(\lambda+(d+2n)/\delta)\Gamma(-(d+2n)/\delta)}{n!\Gamma(n+d/2)}\left(\frac{z}{2}\right)^{2n},$		(14)

Next, we invoke the expression of Fox–Wright function as in (7). In particular, using Equations (8) and (7) we obtain a new form of Dagum spectral density:

	$\displaystyle\widehat{\cal D}_{\delta,\lambda}(\boldsymbol{z})$	$\displaystyle=\delta^{(d)}(\boldsymbol{z})-\frac{\pi^{-d/2}z^{-d}}{\Gamma(\lambda)}\,_{2}\Psi_{1}\Big{[}\begin{matrix}(\lambda,1)&(d/2,-\delta/2)\\ &(0,\delta/2)\end{matrix};-\left(\frac{z}{2}\right)^{\delta}\Big{]}$
		$\displaystyle-\frac{1}{2^{d}\pi^{d/2}}\frac{1}{\Gamma(\lambda)}\frac{2}{\delta}\,_{2}\Psi_{1}\Big{[}\begin{matrix}(\lambda+d/\delta,2/\delta)&(-d/\delta,-2/\delta)\\ &(d/2,1)\end{matrix};-\left(\frac{z}{2}\right)^{2}\Big{]}.$

Theorem 3.4 states that the asymptotic $z\to 0$ for this Fourier transform $\widehat{\cal D}_{\delta,\lambda}(\boldsymbol{z})$ of the Dagum correlation function should be a constant when $\delta>d.$ When we put $z\to 0$ in (14), we may see that only the Dirac $\delta$ function, the $n=0$ term in the first sum of (14) and the $n=0$ term in the second sum of (14) remain non-zero if $\delta>d.$ However, the $n=0$ term in the second sum is the constant

-\frac{1}{2^{d}\pi^{d/2}}\frac{1}{\Gamma(\lambda)}\frac{2}{\delta}\frac{\Gamma(\lambda+d/\delta)\Gamma(-d/\delta)}{\Gamma(d/2)},

(15)

This is the same constant that stands in the formulation of Theorem 3.4 that states the limit $z\to 0$ is smooth for $\delta>d$ and the function $\widehat{\cal D}_{\delta,\lambda}(\boldsymbol{z})$ is continuous at $z=0$ if $\delta>d.$ This means the Dirac $\delta$ function should be canceled with the $n=0$ term in the first sum of (14) and for any $z$ the final form of the spectral density is

	$\displaystyle\widehat{\cal D}_{\delta,\lambda}(\boldsymbol{z})$	$\displaystyle=-\frac{z^{-d}}{\pi^{d/2}}\frac{1}{\Gamma(\lambda)}\sum_{n=1}^{\infty}\frac{(-1)^{n}\ \Gamma(\lambda+n)\Gamma(d/2-n\delta/2)}{n!\Gamma(n\delta/2)}\left(\frac{z}{2}\right)^{n\delta}$		(16)
		$\displaystyle-\frac{1}{2^{d}\pi^{d/2}}\frac{1}{\Gamma(\lambda)}\frac{2}{\delta}\sum_{n=0}^{\infty}\frac{(-1)^{n}\Gamma(\lambda+(d+2n)/\delta)\Gamma(-(d+2n)/\delta)}{n!\Gamma(n+d/2)}\left(\frac{z}{2}\right)^{2n},$		(16)

This may be only if the $n=0$ term in the first sum of (14) is interpreted as the Dirac $\delta$ function in the the sense of distributions.

Proof A.4 (Proof of Theorem 3.4)

The first point can be proved by direct construction. We have

\displaystyle\widehat{\cal D}_{\delta,\lambda}(\boldsymbol{z})

\displaystyle=\frac{z^{1-\frac{d}{2}}}{(2\pi)^{\frac{d}{2}}}\int_{0}^{\infty}J_{\frac{d}{2}-1}(rz)r^{\frac{d}{2}}\left(1-\frac{r^{\delta\lambda}}{(1+r^{\delta})^{\lambda}}\right)\,\mathrm{d}r.

(17)

To find the low frequency behavior of $\widehat{\cal D}_{\delta,\lambda}(\boldsymbol{z})$ , we make use of Equation (17). A change of variable of the type $y=rz$ shows that

	$\displaystyle\widehat{\cal D}_{\delta,\lambda}(\boldsymbol{z})$	$\displaystyle=\frac{z^{-\frac{d}{2}}}{(2\pi)^{\frac{d}{2}}}\int_{0}^{\infty}J_{\frac{d}{2}-1}(y)\left(1-\frac{(y/z)^{\delta\lambda}}{(1+(y/z)^{\delta})^{\lambda}}\right)(y/z)^{d/2}\text{d}y$		(18)
		$\displaystyle=\frac{z^{-d}}{(2\pi)^{\frac{d}{2}}}\int_{0}^{\infty}J_{\frac{d}{2}-1}(y)\left(1-\frac{(y/z)^{\delta\lambda}}{(1+(y/z)^{\delta})^{\lambda}}\right)y^{d/2}\text{d}y$		(18)

We now invoke the identity

\displaystyle 1-\frac{1}{(1+(z/y)^{\delta})^{\lambda}}=-\sum_{j=1}^{\infty}\frac{(-1)^{j}}{j!}\frac{\Gamma(\lambda+j)}{\Gamma(\lambda)}(z/y)^{j\delta}\sim\lambda(z/y)^{\delta},\qquad\hbox{as }z\to 0^{+},

(19)

to obtain

\widehat{\cal D}_{\delta,\lambda}(\boldsymbol{z})\sim\frac{\lambda z^{-d}}{(2\pi)^{\frac{d}{2}}}\int_{0}^{\infty}J_{\frac{d}{2}-1}(y)\left(\frac{z}{y}\right)^{\delta}y^{d/2}\text{d}y=\frac{\lambda z^{\delta-d}}{(2\pi)^{\frac{d}{2}}}\int_{0}^{\infty}J_{\frac{d}{2}-1}(y)y^{\frac{d}{2}-\delta}\text{d}y.

(20)

Using [29, 14,6.651], we find that if $d/2-1/2<\delta<d$

\widehat{\cal D}_{\delta,\lambda}(\boldsymbol{z})\sim\frac{2^{d/2-\delta}\lambda\Gamma(d/2-\delta/2)}{(2\pi)^{\frac{d}{2}}\Gamma(\delta/2)}z^{\delta-d}=\frac{2^{-\delta}\lambda\Gamma(d/2-\delta/2)}{\pi^{\frac{d}{2}}\Gamma(\delta/2)}z^{\delta-d}.

(21)

This result coincides with $n=1$ term of the first sum of Eqs. (14) and (16). We may conclude this limit $z\to 0$ is singular for the Dagum spectral density if $\delta<d.$ The Dagum spectral density is not continuous at the point $z=0$ under the condition $\delta<d.$ The limit $z\to 0$ corresponds to the behaviour of the Dagum correlation function at $r\to\infty$ and to integrability of its Fourier transformation. In this case the Dagum spectral density is singular at $z=0$ because the integral of the Fourier transformation is not convergent for $z=0$ under the condition $\delta<d.$

We now prove the second point. When $z\to 0^{+}$ , the Bessel of the second kind can be expressed asymptotically as

J_{\nu}(rz)\sim\frac{(rz)^{\nu}}{2^{\nu}\Gamma(\nu+1)}.

(22)

Thus we may write in this limit

	$\displaystyle\widehat{\cal D}_{\delta,\lambda}(\boldsymbol{z})=\frac{z^{1-\frac{d}{2}}}{(2\pi)^{\frac{d}{2}}}\int_{0}^{\infty}J_{\frac{d}{2}-1}(rz)r^{\frac{d}{2}}\left(1-\frac{r^{\delta\lambda}}{(1+r^{\delta})^{\lambda}}\right)\text{d}r$
	$\displaystyle\sim\frac{z^{1-\frac{d}{2}}}{(2\pi)^{\frac{d}{2}}}\int_{0}^{\infty}\frac{(rz)^{\frac{d}{2}-1}}{2^{\frac{d}{2}-1}\Gamma(d/2)}r^{\frac{d}{2}}\left(1-\frac{r^{\delta\lambda}}{(1+r^{\delta})^{\lambda}}\right)\text{d}r$
	$\displaystyle=\frac{1}{\pi^{\frac{d}{2}}2^{d-1}\Gamma(d/2)}\int_{0}^{\infty}r^{d-1}\left(1-\frac{r^{\delta\lambda}}{(1+r^{\delta})^{\lambda}}\right)\text{d}r.$

We make a change of variable of the type $r^{\delta}=u$ , and we find that, if $\delta>d$ ,

\displaystyle\widehat{\cal D}_{\delta,\lambda}(\boldsymbol{z})

\displaystyle\sim\frac{1}{\delta\pi^{\frac{d}{2}}2^{d-1}\Gamma(d/2)}\int_{0}^{\infty}u^{(d-\delta)/\delta}\left(1-\frac{u^{\lambda}}{(1+u)^{\lambda}}\right)\text{d}u

(23)

This integral is a finite constant under the condition $\delta>d$ and may be found by the change of the variables

\displaystyle\tau=\frac{u}{1+u}\Rightarrow u=\frac{\tau}{1-\tau}:

	$\displaystyle\int_{0}^{\infty}u^{(d-\delta)/\delta}\left(1-\frac{u^{\lambda}}{(1+u)^{\lambda}}\right)\text{d}u$	$\displaystyle=-\int_{0}^{1}\frac{\tau^{\frac{d}{\delta}-1}}{(1-\tau)^{\frac{d}{\delta}+1}}\left(1-\tau^{\lambda}\right)\text{d}\tau$
		$\displaystyle=-\lim_{\varepsilon\to 0}\int_{0}^{1}\frac{\tau^{\frac{d}{\delta}-1}}{(1-\tau)^{\frac{d}{\delta}+1-\varepsilon}}\left(1-\tau^{\lambda}\right)\text{d}\tau$
		$\displaystyle=-\lim_{\varepsilon\to 0}\left[{\rm B}\left(\frac{d}{\delta},-\frac{d}{\delta}+\varepsilon\right)-{\rm B}\left(\frac{d}{\delta}+\lambda,-\frac{d}{\delta}+\varepsilon\right)\right]$

		$\displaystyle=-\lim_{\varepsilon\to 0}\left[\frac{\Gamma\left(\frac{d}{\delta}\right)}{\Gamma(\varepsilon)}-\frac{\Gamma\left(\frac{d}{\delta}+\lambda\right)}{\Gamma(\varepsilon+\lambda)}\right]\Gamma\left(-\frac{d}{\delta}+\varepsilon\right)$		(24)
		$\displaystyle=\frac{\Gamma\left(\frac{d}{\delta}+\lambda\right)\Gamma\left(-\frac{d}{\delta}\right)}{\Gamma(\lambda)}.$		(24)

Thus, we have

\displaystyle\widehat{\cal D}_{\delta,\lambda}(\boldsymbol{z})\sim\frac{1}{\delta\pi^{\frac{d}{2}}2^{d-1}\Gamma(d/2)}\frac{\Gamma(-\mathrm{d}/\delta)\Gamma(d/\delta+\lambda)}{\Gamma(\lambda)}.

This result coincides with $n=0$ term (15) of the second sum of Eqs. (14) and (16). We may conclude this limit $z\to 0$ is smooth for the Dagum spectral density if $\delta>d.$ The Dagum spectral density is continuous at the point $z=0$ under the condition $\delta>d.$ The limit $z\to 0$ corresponds to the behaviour of the Dagum correlation function at $r\to\infty$ and to integrability of its Fourier transformation. In this case the Dagum spectral density is not singular at $z=0$ because the integral of the Fourier transformation is convergent for $z=0$ under the condition $\delta>d.$

Proof A.5 (Proof of Theorem 3.5)

To find the high frequency behaviour of the Dagum spectral density, we need to use Equation (6). Indeed, as $z\to\infty$ , we have

$\displaystyle\widehat{\cal D}_{\delta,\lambda}(\boldsymbol{z})$	$\displaystyle=-\frac{z^{1-\frac{d}{2}}}{2^{\frac{d}{2}-1}\pi^{\frac{d}{2}+1}}\mathrm{Im}\int_{0}^{\infty}K_{\frac{d}{2}-1}(zt)\left(1-\frac{t^{\delta\lambda}e^{i\pi\delta\lambda/2}}{(1+e^{i\pi\delta/2}t^{\delta})^{\lambda}}\right)t^{\frac{d}{2}}\,\mathrm{d}t$	(25)
	$\displaystyle=-\frac{z^{-d}}{2^{\frac{d}{2}-1}\pi^{\frac{d}{2}+1}}\mathrm{Im}\int_{0}^{\infty}K_{\frac{d}{2}-1}(t)\left(1-\frac{z^{-\delta\lambda}t^{\delta\lambda}e^{i\pi\delta\lambda/2}}{(1+e^{i\pi\delta/2}z^{-\delta}t^{\delta})^{\lambda}}\right)t^{\frac{d}{2}}\,\mathrm{d}t$
	$\displaystyle=\frac{z^{-d}}{2^{\frac{d}{2}-1}\pi^{\frac{d}{2}+1}}\mathrm{Im}\int_{0}^{\infty}K_{\frac{d}{2}-1}(t)(1+z^{\delta}e^{-i\pi\delta/2}t^{-\delta})^{-\lambda}t^{\frac{d}{2}}\,\mathrm{d}t$
	$\displaystyle=\frac{z^{-d}}{2^{\frac{d}{2}-1}\pi^{\frac{d}{2}+1}}\mathrm{Im}\int_{0}^{\infty}K_{\frac{d}{2}-1}(t)z^{-\delta\lambda}e^{i\pi\lambda\delta/2}t^{\lambda\delta}(1+z^{-\delta}e^{i\pi\delta/2}t^{\delta})^{-\lambda}t^{\frac{d}{2}}\,\mathrm{d}t$
	$\displaystyle=\frac{z^{-d}}{2^{\frac{d}{2}-1}\pi^{\frac{d}{2}+1}}\mathrm{Im}\int_{0}^{\infty}K_{\frac{d}{2}-1}(t)z^{-\delta\lambda}e^{i\pi\lambda\delta/2}t^{\lambda\delta}\sum_{k=0}^{\infty}\frac{(-1)^{k}\Gamma(\lambda+k)e^{ik\pi\delta/2}t^{k\delta}}{k!\Gamma(\lambda)z^{k\delta}}t^{\frac{d}{2}}\,\mathrm{d}t$
	$\displaystyle=\frac{z^{-d-\delta\lambda}}{2^{\frac{d}{2}-1}\pi^{\frac{d}{2}+1}}\mathrm{Im}\sum_{k=0}^{\infty}\frac{(-1)^{k}\Gamma(\lambda+k)e^{i(k+\lambda)\pi\delta/2}}{k!\Gamma(\lambda)z^{k\delta}}\int_{0}^{\infty}K_{\frac{d}{2}-1}(t)t^{(k+\lambda)\delta+\frac{d}{2}}\,\mathrm{d}t.$

Using [29, 6.561,16], we obtain

	$\displaystyle\widehat{\cal D}_{\delta,\lambda}(\boldsymbol{z})$	$\displaystyle=\frac{z^{-d-\delta\lambda}}{\pi^{\frac{d}{2}+1}}\mathrm{Im}\sum_{k=0}^{\infty}\frac{(-1)^{k}2^{(k+\lambda)\delta}\Gamma(\lambda+k)\Gamma(\frac{d+(k+\lambda)\delta}{2})\Gamma(1+\frac{(k+\lambda)\delta}{2})e^{i(k+\lambda)\pi\delta/2}}{k!\Gamma(\lambda)z^{k\delta}}$
		$\displaystyle=\frac{z^{-d-\delta\lambda}}{\pi^{\frac{d}{2}+1}}\sum_{k=0}^{\infty}\frac{(-1)^{k}2^{(k+\lambda)\delta}\Gamma(\lambda+k)\Gamma(\frac{d+(k+\lambda)\delta}{2})\Gamma(1+\frac{(k+\lambda)\delta}{2})\sin{(k+\lambda)\pi\delta/2}}{k!\Gamma(\lambda)z^{k\delta}}.$

We use Euler's formula given by

\sin(k\delta\pi/2)=\frac{\pi}{\Gamma(k\delta/2)\Gamma(1-k\delta/2)},

and we write

	$\displaystyle\widehat{\cal D}_{\delta,\lambda}(\boldsymbol{z})$	$\displaystyle=\frac{2^{\delta\lambda}z^{-d-\delta\lambda}}{\pi^{\frac{d}{2}}\Gamma(\lambda)}\sum_{k=0}^{\infty}\frac{(-1)^{k}\Gamma(\lambda+k)\Gamma(d/2+(k+\lambda)\delta/2)\Gamma(1+(k+\lambda)\delta/2)}{k!\Gamma((k+\lambda)\delta/2)\Gamma(1-(k+\lambda)\delta/2)}\left(\frac{2}{z}\right)^{k\delta}$		(26)
		$\displaystyle=\frac{2^{\delta\lambda}z^{-d-\delta\lambda}}{\pi^{\frac{d}{2}}\Gamma(\lambda)}\,_{3}\Psi_{2}\Big{[}\begin{matrix}(\lambda,1)&(\delta\lambda/2+1,\delta/2)&((\delta\lambda+d)/2,\delta/2)\\ &(\delta\lambda/2,\delta/2)&(1-\delta\lambda/2,-\delta/2)\end{matrix};-\left(\frac{2}{z}\right)^{\delta}\Big{]}.$		(26)

However, in this limit we should leave only the first term of this sum

\displaystyle\widehat{\cal D}_{\delta,\lambda}(\boldsymbol{z})

\displaystyle\sim\frac{2^{\delta\lambda-1}\lambda\delta z^{-d-\delta\lambda}}{\pi^{\frac{d}{2}}}\frac{\Gamma(d/2+\lambda\delta/2)}{\Gamma(1-\lambda\delta/2)}

(27)

The proof is completed.

At the end, we would like to comment that the same result for the asymptotic (27) of the spectral density may be found by transforming the integrand of the contour integral (13) to another integrand such that the convergence of the series which arises in the result of the residues calculus when we close the contour of the integral to the left complex infinity becomes obvious because this series would satisfy the standard criteria of convergence. Such a form of the integrand is useful to study the asymptotic of the spectral density at $z\to\infty$ and would allow us to reproduce the asymptotical result (27).

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	$\displaystyle\int_{0}^{\pi/2}e^{-i\delta\varphi j}e^{i\|z\|\xi e^{i\varphi}}\,\mathrm{d}\varphi=$	$\displaystyle\frac{1}{i}\int_{0}^{\pi/2}e^{-i(j\delta+1)\varphi}e^{i\|z\|\xi e^{i\varphi}}\,\mathrm{d}e^{i\varphi}$
	$\displaystyle=$	$\displaystyle\frac{1}{i}\int_{C_{1}}e^{i\|z\|\xi\omega}\omega^{-(j\delta+1)}\,\mathrm{d}\omega=\frac{(\|z\|\xi)^{j\delta}}{i}\int_{C_{\|z\|\xi}}e^{iu}u^{-(j\delta+1)}\,\mathrm{d}u,$

A Deep Look into the Dagum Family of Isotropic Covariance Functions