1 Introduction

The class of continuous covariance functions is completely characterized by Bochner’s theorem: this important result shows that a covariance function (c.f.) is, in general, a complex valued function; for this reason, the whole family of real c.f.s, which are often utilized in several applications (Cressie and Huang 1999; De Iaco et al. 2001, 2002; Gneiting 2002b), only represents a subset of the wider set of complex c.f.s and are, very often, positive in the whole set in which they are defined.

Properties of c.f.s for second order stationary random fields are well known in the literature (Yaglom 1987; Christakos 2017); in particular, it is well known that, in general, the difference of two c.f.s is not a c.f.

In the literature, a few authors have addressed the problem of investigating the difference (or linear combinations with negative weights) of two c.f.s; moreover, this kind of analysis has been performed only in the real domain. At this purpose, a linear combination, with negative weights, of two real valued spatial or spatio-temporal c.f.s has been analyzed by Ma (2005): however, the two real c.f.s involved in the analysis are isotropic and belong to the same family. In addition, a class of c.f.s which allows negative values has also been proposed by Gregori et al. (2008): although it is not required that the c.f.s involved belong to the same class, the applications are however restricted to the case of Gaussian and Matern families (Matern 1980).

In the very last years some efforts have been made in geostatistics utilizing complex formalism. In particular, an application to predict a wind field has been proposed by De Iaco and Posa (2016), and a first attempt of complex formalism in the spatio-temporal context has been provided by Cappello et al. (2020).

As far as we know, there are no contributions in the literature concerning the difference of two c.f.s in the complex domain. Hence, this paper can be considered one of the first attempts to explore the conditions under which the difference of two c.f.s in the complex domain is again a c.f.: these results can be added to the classical properties of c.f.s, well known in the literature. This paper thus yields a general contribution to the theory of correlation, hence the results cannot be restricted to the particular field of application (i.e., geostatistics, time series analysis). In particular, valid models for the difference of c.f.s in the complex domain as well as in the real domain will be proposed; moreover, in order to provide a complete scenario of the subject to be utilized in several and different situations, some examples of the difference of two isotropic c.f.s are also explored for the two and three dimensional Euclidean spaces.

As will be shown, if the conditions for which the difference of two c.f.s provides a new c.f., in \({\mathbb {R}}\), are satisfied, the models obtained through this difference are characterized by different features: for example, they could be non negative in the whole domain, or could be characterized by negative values in a subset of their domain. Indeed, in several applications concerning biology, hydrology and spatio-temporal turbolence, c.f.s with negative values are often needed (Shkarofsky 1968; Pomeroy et al. 2003).

The various examples, concerning the difference of two c.f.s and proposed in this paper, are flexible enough to be utilized in several applications pertaining to the complex domain, as well as to the subset of the real domain. In particular, it will be shown how some special families of c.f.s, obtained through the difference of two c.f.s, present some relevant and flexible characteristics which the standard parametric families of c.f.s and their linear combinations with non negative coefficients, as well as their products, are not able to satisfy.

This paper is organized as follows: in Sect. 2 some characteristics of continuous c.f.s in the complex domain are summarized. In Sect. 3 conditions for which the difference of two c.f.s is again a c.f. have been analyzed in the complex domain, as well as in the special case of the real domain through a detailed analysis involving the parameters of these models; in particular, starting from separable models, the c.f.s obtained through the difference are non separable. In the case of real c.f.s, some well known models, often utilized in the applications, have been considered; some examples of the difference of two isotropic c.f.s are also explored. The results can be utilized in a flexible way in any dimensional domain. An overview and some relevant hints about how to construct c.f.s with negative values have been given in Sect. 4, whereas the relevance of the results has been pointed out in Sect. 5.

2 A brief overview

In this section, a brief outline of continuous c.f.s, which are completely characterized by Bochner’s theorem (Bochner 1959), is provided. Let’s denote with \({\mathbb {R}}^n\) the Euclidean n-dimensional space, with \({\mathbb {C}}\) the set of the complex numbers and i the imaginary unit.

Theorem 1

Bochner’s theorem. Let \(C: {\mathbb {R}}^n \rightarrow {\mathbb {C}}\), be a Hermitian function, then C is a continuous c.f. if and only if it is of the form

$$\begin{aligned} C(\mathbf{x}) = \int _{{\mathbb {R}}^n} \exp ({i{\varvec{\omega }}^T \mathbf{x}}) dF({\varvec{\omega }}), \end{aligned}$$
(1)

where F is a non decreasing and non negative bounded measure on \({\mathbb {R}}^n\).

The c.f. C in (1) can also be written as: \(\quad C(\mathbf{x}) = C_{re}+i C_{im},\) where

$$C_{re}=\int _{{\mathbb {R}}^n} \cos ({{\varvec{\omega }}^T \mathbf{x}}) dF({\varvec{\omega }})$$

and

$$C_{im}= \int _{{\mathbb {R}}^n} sen({{\varvec{\omega }}^T \mathbf{x}}) dF({\varvec{\omega }});$$

in particular, \(C_{re}\) is a real c.f., whereas \(C_{im}\) is an odd function and it is not a c.f..

According to the previous theorem, any continuous c.f. can be represented as in (1) and the converse is also true. Please note that as it is the Fourier transform of the finite measure F, the function C must be continuous. From (1) it follows that:

  1. 1.

    \(C(\mathbf{0})\ge 0\);

  2. 2.

    \(C(-\mathbf{x})=\overline{C}(\mathbf{x})\);

  3. 3.

    \(|C(\mathbf{x})|\le C(\mathbf{0}),\)

where the bar denotes complex conjugate. If the non negative spectral distribution function F is absolutely continuous, then there exists: \(f : {\mathbb {R}}^n \longrightarrow {\mathbb {R}},\) such that:

$$\begin{aligned} C(\mathbf{x}) = \int _{{\mathbb {R}}^n} \exp ({i{\varvec{\omega }}^T\mathbf{x}}) f({\varvec{\omega }})d{\varvec{\omega }}, \end{aligned}$$
(2)

where f is called spectral density function, it is non negative and integrable. In general, there is no guarantee that C is integrable. If C is integrable, then the spectral distribution function F is absolutely continuous and the spectral density function f can be expressed in terms of the c.f. C, i.e.,

$$\begin{aligned} f({\varvec{\omega }})={1\over {(2\pi )^n}}\int _{{\mathbb {R}}^n} \exp (-i{\varvec{\omega }}^T \mathbf{x} ) C(\mathbf{x} )d\mathbf{x} . \end{aligned}$$
(3)

If F is an absolutely continuous spectral distribution function, then

$$\begin{aligned} \tilde{F}({\varvec{\omega }})=C(\mathbf{0} )-F(-{\varvec{\omega }}) \end{aligned}$$

is also a an absolutely continuous spectral distribution function and it is called the conjugate spectral distribution of F.

Corollary 1

The c.f. of the conjugate spectral distribution \(\tilde{F}\) is the conjugate c.f. of C, i.e.,

$$\begin{aligned} \overline{C} (\mathbf{x}) = \int _{{\mathbb {R}}^n} \exp ({i{\varvec{\omega }}^T \mathbf{x}}) d\tilde{F}({\varvec{\omega }}). \end{aligned}$$

Hence, if C is a c.f., then \(\overline{C}\) is also a c.f..

Corollary 2

If the spectral distribution function F is absolutely continuous, then the c.f. of the spectral density function \(\tilde{f}({\varvec{\omega }})=f(-{\varvec{\omega }})\), corresponding to the conjugate spectral distribution \(\tilde{F}\), is the conjugate c.f. of C of the spectral density function \(\tilde{f}\), i.e.,

$$\begin{aligned} \overline{C}(\mathbf{x}) = \int _{{\mathbb {R}}^n} \exp ({i{\varvec{\omega }}^T \mathbf{x}}) \tilde{f}({\varvec{\omega }}) d({\varvec{\omega }}). \end{aligned}$$

An absolutely continuous spectral distribution function is said to be symmetric if it is equal to its conjugate, i.e., \(\tilde{F}({\varvec{\omega }})=F({\varvec{\omega }})\). In this last case, the spectral density function is an even function and the corresponding covariance is a real function. Equivalently, a c.f. is a real function if and only if it is equal to its conjugate, i.e., \(\overline{C}(\mathbf{x})=C(\mathbf{x}).\) In order to construct models for the difference of two c.f.s, it is worth underlining the following aspects:

  • if the spectral distribution function is not symmetric or, alternatively, the spectral density function is not an even function, the c.f. will necessarily be a complex c.f..

  • On the other hand, if the spectral distribution function is symmetric or, alternatively, the spectral density function is an even function, the c.f. will necessarily be a real c.f.. In this case, the two spectral density functions involved in the difference of two c.f.s, are both even functions, or, as will be shown in the next section, the two spectral density functions couldn’t both be even functions, however their difference could be, in some cases, an even function.

  • Complex c.f.s can be constructed in any dimensional space; at this purpose, it is important to provide some useful details about the various techniques which have been proposed in the literature. In particular,

    • utilizing the well known closure properties of c.f.s;

    • through the construction of separable models, i.e.

      $$\begin{aligned} C(\mathbf{x} )=\prod _{i=1}^n C_i(x_i), \quad \mathbf{x} =(x_1,\ldots ,x_n) \in {\mathbb {R}}^n, \end{aligned}$$
      (4)

      where each component in the product is a c.f. in \({\mathbb {R}}\); in particular, if \(F_i\) is the spectral distribution function of \(C_i\) in the representation (1), \(i=1,\ldots ,n\), then C also has a representation as in (1), where

      $$\begin{aligned} F(\varvec{\omega })=\prod _{i=1}^n F_i(\omega _i), \quad \varvec{\omega }=(\omega _1,\ldots ,\omega _n)\in {\mathbb {R}}^n; \end{aligned}$$
      (5)

    • by considering the subclass of isotropic c.f.s; in this case, different results based on the spectral representation through Bessel functions (Matern 1980; Yaglom 1987), completely monotone functions (Schoenberg 1938), sufficient conditions on positive definiteness for isotropic c.f.s (Polya 1949), have been utilized.

3 Difference of two covariance functions

As already underlined, the results given by Ma (2005) and Gregori et al. (2008), concerning the difference between two c.f.s, are only valid in the real domain. On the other hand, the corollaries presented in this section are valid in the complex domain, i.e., for any c.f. defined as in (1); hence, these results can be considered further properties for the difference of two c.f.s in the complex domain. Valid models for the difference of c.f.s in the complex domain, as well as in the real domain, will be constructed by utilizing the results of the same corollaries. The dependence of a c.f. from a vector of parameters will be properly specified, because all the results depend on these parameters.

Corollary 3

Let \(C_i: {\mathbb {R}}^n \rightarrow {\mathbb {C}}, i=1,2\) be c.f.s and define

$$\begin{aligned} C(\mathbf{x} ; {\varvec{\Lambda }})=A C_1(\mathbf{x} ; \varvec{\alpha })-B C_2(\mathbf{x} ; \varvec{\beta }), \quad \mathbf{x} =(x_1,\ldots ,x_n)\in {{\mathbb {R}}^n}, \end{aligned}$$
(6)

where \(A>0, B>0, \varvec{\alpha }\) and \(\varvec{\beta }\) are vectors of parameters and \({\varvec{\Lambda }}=(A,B, {\varvec{\alpha }}, {\varvec{\beta }}).\)

Because of Bochner’s theorem, two different cases need to be analyzed:

  • the spectral distribution functions \(F_i, i=1,2\), which define \(C_i, i=1,2\), respectively, are not absolutely continuous. In this case the corresponding c.f.s can be written as follows:

    $$\begin{aligned}&C_1(\mathbf{x} ; \varvec{\alpha })=\int _{{\mathbb {R}}^n} \exp (i{\varvec{\omega }}^T \mathbf{x} ) dF_1({\varvec{\omega }; \varvec{\alpha }}), \\&C_2(\mathbf{x} ; \varvec{\beta })=\int _{{\mathbb {R}}^n} \exp (i{\varvec{\omega }}^T \mathbf{x} ) dF_2({\varvec{\omega }; \varvec{\beta }}); \end{aligned}$$

    then

    $$\begin{aligned} C(\mathbf{x} ; {\varvec{\Lambda }})=\int _{{\mathbb {R}}^n} \exp (i{\varvec{\omega }}^T \mathbf{x} ) d( A F_1({\varvec{\omega };\varvec{\alpha }})- B F_2({\varvec{\omega };\varvec{\beta }})) \end{aligned}$$
    (7)

    is a c.f. if and only if, for some suitable values of the vector of parameters \(\varvec{\Lambda }\) and \({\varvec{\omega }} \in {\mathbb {R}}^n\), \((A F_1- B F_2)\) is a non negative bounded function on \({\mathbb {R}}^n\);

  • the spectral distribution functions \(F_i, i=1,2\), which define \(C_i, i=1,2\), respectively, are absolutely continuous. In this case, the corresponding c.f.s can be written as follows:

    $$\begin{aligned}&C_1(\mathbf{x} ; \varvec{\alpha })=\int _{{\mathbb {R}}^n} \exp (i{\varvec{\omega }}^T \mathbf{x} ) f_1({\varvec{\omega }; \varvec{\alpha }})d{\varvec{\omega }}, \\&C_2(\mathbf{x} ; \varvec{\beta })=\int _{{\mathbb {R}}^n} \exp (i{\varvec{\omega }}^T \mathbf{x} ) f_2({\varvec{\omega }; \varvec{\beta }})d{\varvec{\omega }}, \end{aligned}$$

    with \(f_i({\varvec{\omega }; \cdot })\ge 0, {\varvec{\omega }} \in {\mathbb {R}}^n\) and \(\int _{{\mathbb {R}}^n} f_i({\varvec{\omega }; \cdot })d{\varvec{\omega }} <\infty , i=1,2;\) then

    $$\begin{aligned} C(\mathbf{x} ; {\varvec{\Lambda }})=\int _{{\mathbb {R}}^n} \exp (i{\varvec{\omega }}^T \mathbf{x} )( A f_1({\varvec{\omega };\varvec{\alpha }})- B f_2({\varvec{\omega };\varvec{\beta }})) d{\varvec{\omega }} \end{aligned}$$
    (8)

    is a c.f. if and only if, for some suitable values of the vector of parameters \(\varvec{\Lambda }\) and \({\varvec{\omega }} \in {\mathbb {R}}^n,\)

    $$\begin{aligned}&A f_1({\varvec{\omega }; \varvec{\alpha }})- B f_2({\varvec{\omega }; \varvec{\beta }})\ge 0, \nonumber \\&\quad \int _{{\mathbb {R}}^n} (A f_1({\varvec{\omega }; \varvec{\alpha }})- B f_2({\varvec{\omega };\varvec{\beta }})) d{\varvec{\omega }}<+\infty . \end{aligned}$$
    (9)

The second property of (9) is always satisfied because \((A f_1 - B f_2)\) is a difference of integrable functions, hence only the first property of (9) need to be verified.

Two special cases of Corollary 3 are given hereafter. In particular, in the next Corollary 4 it is assumed that the function C in (6) is defined as the product of the difference of c.f.s, where each c.f. in the difference is defined in \({\mathbb {R}}\), whereas, in the next Corollary 5 it is assumed that the function C in (6) is defined as the difference of the product of c.f.s, where each c.f. is defined in \({\mathbb {R}}.\)

Of course, the above two models are completely different; in particular, the model in the Corollary 4 has been built by considering the following aspects:

  1. 1.

    it is a special case of Corollary 3 in \({\mathbb {R}}\),

  2. 2.

    it is an application of Eq. (4).

In the Corollary 5, starting from separable models in \({\mathbb {C}}\), the components of the real and imaginary part of the resulting c.f. are non separable and anisotropic.

Corollary 4

Let \(C_{j}: {\mathbb {R}} \rightarrow {\mathbb {C}}, j=1,2\) be c.f.s and define

$$\begin{aligned}&C_{12(k)}(x_k;\alpha _k,\beta _k)=A_k C_1(x_k;\alpha _k)\nonumber \\&\quad -B_k C_2(x_k; \beta _k), A_k>0, B_k>0, k=1,\ldots ,n, \end{aligned}$$
(10)

and

$$\begin{aligned} C(\mathbf{x} ; \varvec{\Theta })= \prod _{k=1}^n \bigg [A_k C_{1}(x_k;\alpha _k)-B_k C_2(x_k;\beta _k)\bigg ], \end{aligned}$$
(11)

where \(\mathbf{A} =(A_1,\ldots ,A_n), \mathbf{B} =(B_1,\ldots ,B_n), \varvec{\alpha }=(\alpha _1,\ldots ,\alpha _n)\) and \(\varvec{\beta }=(\beta _1,\ldots ,\beta _n)\) are vectors of parameters, \(\mathbf{x} =(x_1, \ldots , x_n)\in {\mathbb {R}}^n\) and \({\varvec{\Theta }}=(\mathbf{A} , \mathbf{B} , {\varvec{\alpha }}, {\varvec{\beta }}).\)

Because of Bochner’s theorem, two different cases need to be analyzed:

  • the spectral distribution functions \(F_1(x_k;\alpha _k), F_2(x_k;\beta _k)\), which define \(C_1(x_k;\alpha _k)\) and \(C_2(x_k;\beta _k), k=1,\ldots ,n,\) respectively, are not absolutely continuous. In this case the c.f.s can be written as follows:

    $$\begin{aligned}&C_1(x_k; \alpha _k)=\int _{{\mathbb {R}}} \exp (i{\omega _k} {x_k}) dF_1({\omega _k; \alpha _k}), \\&C_2({x_k}; \beta _k)=\int _{{\mathbb {R}}} \exp (i{\omega _k} {x_k}) dF_2({\omega _k;\beta _k}); \end{aligned}$$

    then, each

    $$\begin{aligned} C_{12(k)}({x_k}; \alpha _k, \beta _k)=\int _{{\mathbb {R}}} \exp (i{\omega _k} {x_k}) d( A_k F_k({\omega _k;\alpha _k})- B_k F_2({\omega _k; \beta _k})), \end{aligned}$$
    (12)

    is a c.f. if and only if, for some suitable values of the vector of parameters \(\varvec{\Theta }\) and \({\omega _k} \in {\mathbb {R}}, k=1,\ldots ,n\)

    $$\begin{aligned} A_k F_1({\omega _k;\alpha _k})- B_k F_2({\omega _k;\beta _k})\ge 0, \quad k=1,\ldots ,n, \end{aligned}$$
    (13)

    and each \((A_k F_1-B_k F_2), k=1,\ldots ,n,\) is a non decreasing function.

    As a consequence, C defined in Eq. (11) is a c.f..

  • The spectral distribution functions \(F_1(x_k;\alpha _k), F_2(x_k;\beta _k)\), which define \(C_1(x_k;\alpha _k)\) and \(C_2(x_k;\beta _k), k=1,\ldots ,n,\) respectively, are absolutely continuous. In this case, these c.f.s can be written as follows:

    $$\begin{aligned} C_1({x_k}; \alpha _k)& = \int _{{\mathbb {R}}} \exp (i{\omega _k} {x_k}) f_1({ \omega _k; \alpha _k})d{ \omega _k},\\ C_2({x_k}; \beta _k)& = \int _{{\mathbb {R}}} \exp (i{ \omega _k} {x_k}) f_2({\omega _k; \beta _k})d{\omega _k}, \end{aligned}$$

    with \(f_j({ \omega _k; \cdot })> 0, {\omega _k} \in {\mathbb {R}}\) and \(\int _{{\mathbb {R}}} f_j({\omega _k; \cdot })d{\omega _k}<\infty, \quad j=1,2; k=1,\ldots ,n;\)

    then, each

    $$\begin{aligned}&C_{12(k)}({x_k}; \alpha _k, \beta _k)\\=\int _{{\mathbb {R}}} \exp (i{\omega _k} {x_k})d ( A_k F_k({\omega _k;\alpha _k})\nonumber \quad - B_k F_2({\omega _k; \beta _k})) \end{aligned}$$
    (14)

    is a c.f. if and only if, for some suitable values of the vector of parameters \(\varvec{\Theta }\) and \({\omega _k} \in {\mathbb {R}}, k=1,\ldots ,n,\)

    $$\begin{aligned}&A_k f_1({\omega _k; \alpha _k})- B_k f_2({\omega _k; \beta _k})\ge 0, \nonumber \\&\quad \int _{{\mathbb {R}}} (A_k f_1({\omega _k; \alpha _k})- B_k f_2({\omega _k; \beta _k})) d{\omega _k}<+\infty . \end{aligned}$$
    (15)

    Note that the second condition in (15) is always satisfied; as a consequence of Eq. (4), C defined in Eq. (11) is a c.f..

Corollary 5

Let \(C_{j}: {\mathbb {R}} \rightarrow {\mathbb {C}}, j=1,2\) be c.f.s and define

$$\begin{aligned}&C(\mathbf{x} ; {\varvec{\Theta }})= \prod _{k=1}^n A_k C_1(x_k;\alpha _k)- \prod _{k=1}^n B_k C_2(x_k;\beta _k),\nonumber \\&\quad A_k>0, B_k>0, k=1,\ldots ,n, \end{aligned}$$
(16)

where \(\mathbf{A} =(A_1,\ldots ,A_n), \mathbf{B} =(B_1,\ldots ,B_n), \varvec{\alpha }=(\alpha _1,\ldots ,\alpha _n)\) and \(\varvec{\beta }=(\beta _1,\ldots ,\beta _n)\) are vectors of parameters, \(\mathbf{x} =(x_1, \ldots , x_n)\in {\mathbb {R}}^n\) and \({\varvec{\Theta }}=(\mathbf{A} , \mathbf{B} , {\varvec{\alpha }}, {\varvec{\beta }}).\) Let the spectral distribution functions which define \(C_1(x_k;\alpha _k)\) and \(C_2(x_k;\beta _k), k=1,\ldots ,n,\) respectively, be absolutely continuous, with

$$\begin{aligned}&f_j({ \omega _k; \cdot })> 0, {\omega _k} \in {\mathbb {R}}, \quad \int _{{\mathbb {R}}} f_j({ \omega _k; \cdot })d{\omega _k} <\infty , \\&\quad j=1,2, \quad k=1,\ldots ,n. \end{aligned}$$

In this case, the expression (16) can be written as follows:

$$\begin{aligned} C(\mathbf{x} ; {\varvec{\Theta }})& = \prod _{k=1}^n \bigg [A_k \int _{{\mathbb {R}}} \exp (i\omega _k x_k) f_1({\omega _k; \alpha _k}) d\omega _k\bigg ]\nonumber -\prod _{k=1}^n \bigg [B_k \int _{{\mathbb {R}}} \exp (i\omega _k x_k)f_2({\omega _k; \beta _k}) d\omega _k\bigg ]\nonumber = \int _{{\mathbb {R}}^n} \exp (i \varvec{\omega }^T \mathbf{x} )\nonumber\; \bigg [ \prod _{k=1}^n A_k f_1({\omega _k; \alpha _k})- \prod _{k=1}^n B_k f_2({\omega _k; \beta _k})\bigg ] d\varvec{\omega }. \end{aligned}$$
(17)

Then, (17) is a c.f. if and only if, for some suitable values of the vector of parameters \(\varvec{\Theta }\) and \(\varvec{\omega } \in {\mathbb {R}}^n,\)

$$\begin{aligned} \prod _{k=1}^n \bigg [{A_k f_1({\omega _k; \alpha _k})\over {B_k f_2({\omega _k; \beta _k})}}\bigg ] \ge 1, \quad {\omega _k} \in {\mathbb {R}}, k=1,\ldots ,n. \end{aligned}$$
(18)

Remarks

  • Note that if (15) is satisfied, then inequality (18) is verified; hence, the second part of Corollary 4 implies Corollary 5.

  • No hypothesis on the spectral distribution function (i.e. to be symmetric or not), as well as on the spectral density function (i.e. to be even or not) are made in the Corollaries 3, 4 and 5; hence, these corollaries provide a general result on the c.f. C in (7), (8), (11) and (16), which could be, in general, a complex c.f..

  • The complex c.f. C defined in (11) is the product of complex c.f.s \(C_{12(k)}, k=1,\ldots ,n\); then C is a separable c.f., where separability is referred to the complex domain \({\mathbb {C}}\). However, the real and the imaginary components outcoming from the above product are non separable in \({\mathbb {R}}^n\) and are characterized by non geometric anisotropy.

  • The complex c.f. C defined in (16) is the difference of two products of complex c.f.s, where each product is separable in the complex domain \({\mathbb {C}}\). However, the real and the imaginary components of the resulting complex c.f. are non separable in \({\mathbb {R}}^n\) and are characterized by non geometric anisotropy.

  • If the two spectral density functions \(f_1\) and \(f_2\) in (8) are even functions and conditions (9) are satisfied, then \((A C_1- B C_2)\) is a real c.f..

  • If just one of the two spectral density functions \(f_1\) or \(f_2\) in (8) is an even function, then \((A f_1- B f_2)\) cannot be an even function, as a consequence, if conditions (9) are satisfied, \((A C_1- B C_2)\) is a complex c.f..

  • It may happen that if the two spectral density functions \(f_1\) and \(f_2\) in (8) are not even functions, then \((A f_1- B f_2)\) could be an even function, as a consequence, if conditions (9) are satisfied, \((A C_1- B C_2)\) is a real c.f., as shown in the following example.

Example 1

Consider, in \({\mathbb {R}}\), the following spectral density functions:

$$\begin{aligned} f_1(x)=e^{-|x|} (x^2+x+3), \qquad f_2(x)=e^{-|x|} (x^2+x+2), \end{aligned}$$

which are not even functions, hence the corresponding c.f.s are complex functions. It is easy to verify that: \(f_1(x)-f_2(x)=e^{-|x|},\) is an even spectral density function, as a consequence, the corresponding c.f. is a real function. A generalization of this example will be given in Corollary 7.

  • If \(C_1\) and \(C_2\) are real and continuous c.f.s and \(C_1(\mathbf{x} )\le C_2(\mathbf{x}) , \mathbf{x} \in {\mathbb {R}}^n,\) then \((C_2-C_1)\) is not, in general, a c.f., as shown in the following example.

Example 2

Consider the following c.f.s: \(C_1(x)=e^{-2|x|}, \quad C_2(x)=e^{-|x|},\)with \(C_1(x)\le C_2(x), x \in {\mathbb {R}}\). Note that \(C(x)=C_2(x)-C_1(x)\) is not a c.f., because \(C(0)=0\), and \(C(x)>C(0), x\ne 0\), which contradicts the third property of a c.f., given in Section 2.

  • If \(C_1\) and \(C_2\) are real correlation functions, i.e., \(C_1(\mathbf{0} )=C_2(\mathbf{0} )=1,\) then \(C(\mathbf{x} )=C_1(\mathbf{x} )-C_2(\mathbf{x} )\) cannot ever be a c.f. (except for the trivial case \(C_1(\mathbf{x} )=C_2(\mathbf{x} )=0, \mathbf{x} \in {\mathbb {R}}^n\)); indeed, in this case \(C(\mathbf{0} )=0\), however, \(|C(\mathbf{x} )|\ge C(\mathbf{0} ),\) which contradicts one of the main properties of c.f.s.

  • If the spectral distribution functions \(F_i, i=1,2\), which define \(C_i, i=1,2\), respectively, in Corollary 3 are absolutely continuous, then the resulting c.f. C is strictly positive definite (De Iaco et al. 2011; De Iaco and Posa 2018).

Some significant examples, utilizing the results of the previous corollaries, will be given hereafter. In particular,

  1. 1.

    in the Sect. 3.1, some examples for the difference of c.f.s in the complex domain will be provided.

  2. 2.

    In the Sect. 3.2, some examples for the difference of c.f.s in the real domain will be also given; in this last case, differently from the models described in Ma (2005) which are isotropic and they belong to the same class, the c.f.s proposed in this paper, obtained as the difference of two real c.f.s, are anisotropic and non separable, although they have been built starting from separable models and they do not necessarily belong to the same class. Some well known models, utilized in the applications, have been considered.

  3. 3.

    In the Sect. 3.3 some examples for the difference of two isotropic c.f.s are provided for the two and three dimensional Euclidean space; in particular, for some of these examples the c.f.s involved in the difference could not belong to the same family.

3.1 Difference of two complex covariance functions

The following result shows a special case in which the difference of two spectral distribution functions, which are not absolutely continuous, is again a spectral distribution function. As a consequence, the Fourier transform of this last spectral distribution function, corresponding to the difference of two c.f.s, will be a c.f..

Corollary 6

Let \(F_1\) and \(F_2\) be spectral distribution functions on \({\mathbb {R}},\) i.e.,:

$$\begin{aligned} F_1(\omega ; \varvec{\alpha })& = \left\{ \begin{array}{ll} 0 &{} \omega< t_1 \\ \alpha _ k &{} t_k \le \omega < t_{k+1}\quad i=1,\ldots , n-1\\ \alpha _{n} &{} \omega \ge t_n,\\ \end{array} \right. \end{aligned}$$
(19)
$$\begin{aligned} F_2(\omega ; \varvec{\beta })& = \left\{ \begin{array}{ll} 0 &{} \omega< t_1 \\ \beta _ k &{} t_k \le \omega < t_{k+1}\quad k=1,\ldots , n-1\\ \beta _{n} &{} \omega \ge t_n,\\ \end{array} \right. \end{aligned}$$
(20)

where \(\varvec{\alpha }=(\alpha _1,\ldots ,\alpha _n), \varvec{\beta }=(\beta _1,\ldots ,\beta _n), \quad \beta _k<\alpha _k, \alpha _k>0, \beta _k>0, k=1,\ldots ,n\)with \(\alpha _k<\alpha _{k+1}, \beta _k<\beta _{k+1},\) and \((\alpha _{k}-\beta _{k})<(\alpha _{k+1}-\beta _{k+1}), k=1,\ldots ,n-1;\) let

$$\begin{aligned}&C_1({x};\varvec{\alpha })=\int _{{\mathbb {R}}} \exp (i{ \omega } {x}) dF_1({\omega ;\varvec{\alpha }}), \\&C_2({x};\varvec{\beta })=\int _{{\mathbb {R}}} \exp (i{\omega } {x}) dF_2({\omega ;\varvec{\beta }}). \end{aligned}$$

Then: \(\quad C_1({x};\varvec{\alpha })- C_2({x};\varvec{\beta })=\int _{{\mathbb {R}}} \exp (i{\omega } {x}) d(F_1({\omega ;\varvec{\alpha }})- F_2({\omega ;\varvec{\beta }})),\) is a c.f., because \((F_1-F_2)\) is a spectral distribution function, i.e., it is non negative and a non decreasing function. Moreover, if \((F_1-F_2)\) is a non symmetric spectral distribution function, then \((C_1-C_2)\) is a complex c.f..

Corollary 6 is valid in \({\mathbb {R}}\), however, utilizing Eqs. (4) and (5), it can be extended to any dimensional space \({\mathbb {R}}^n.\) In the next examples, as well as in the examples shown in the Sect. 3.2, the same formalism utilized for the Corollaries 3, 4 and 5 will be retained: the generic coordinate of the one dimensional space \({\mathbb {R}}\) will be denoted with \(x_k, k=1,\ldots ,n\). In this way, starting from the one dimensional space, covariance models in \({\mathbb {R}}^n\) will be easily constructed applying (11) and (16). These aspects will be underlined at the end of the examples shown hereafter and at the end of the examples discussed in the Sect. 3.2.

An application of Corollary 3 for absolutely continuous spectral distribution functions is given in the following examples, where the difference of two spectral density functions is considered: such a difference is a spectral density function if conditions (9) are satisfied; as a consequence, the difference of the corresponding c.f.s is again a c.f.. In particular, in the Examples 3, 4 and 5 the spectral density functions \(f_1\) and \(f_2\) are not both even functions, whereas in the Example 6\(f_1\) is an even function while \(f_2\) is not an even function.

Example 3

Let’s consider, in \({\mathbb {R}}\), the following spectral density functions, with \(0<\beta _k<\alpha _k, 0<B_k<A_k, \omega _k \in {\mathbb {R}}, k=1,\ldots ,n\):

$$\begin{aligned} f_1(\omega _k; \alpha _k)& = \left\{ \begin{array}{ll} 0 &{} \omega _k<0\\ 1 &{} 0\le \omega _k<\alpha _k\\ 0 &{} \omega _k\ge \alpha _k\\ \end{array} \right. \end{aligned}$$
(21)
$$\begin{aligned} f_2(\omega _k; \beta _k)& = \left\{ \begin{array}{ll} 0 &{} \omega _k<0\\ 1 &{} 0\le \omega _k<\beta _k\\ 0 &{} \omega _k\ge \beta _k\\ \end{array} \right. \end{aligned}$$
(22)

which are not even functions; since: \(A_k f_1({\omega _k; \alpha _k})- B_k f_2({\omega _k; \beta _k})\ge 0, \omega _k \in {\mathbb {R}},\) then the following difference:

$$\begin{aligned}&A_k C_1({x_k}; \alpha _k)- B_k C_2({x_k}; \beta _k)\\&\quad =\int _{{\mathbb {R}}} \exp (i{\omega _k} {x_k})( A_k f_1( \omega _k;\alpha _k)- B_k f_2({\omega _k; \beta _k})) d{\omega _k}\\&\quad =A_k {{ \sin (\alpha _k x_k)}\over {x_k}} - B_k {{\sin (\beta _k x_k)}\over {x_k}} \\&\qquad + i \Bigg [ A_k {2\over x_k} {\sin ^2{\bigg ({{\alpha _k x_k}\over 2}\bigg )}} - B_k {2\over x_k} {\sin ^2{\bigg ({{\beta _k x_k}\over 2}\bigg )}}\Bigg ], \end{aligned}$$

with \(A_k C_1({0}; \alpha _k)- B_k C_2({0}; \beta _k)=A_k\alpha _k-B_k\beta _k\), is a complex c.f., \(x_k \in {\mathbb {R}}, k=1,\ldots ,n.\)

According to Corollaries 4 and 5, the following functions:

$$\begin{aligned} C(\mathbf{x} ; \varvec{\Theta })& = \prod _{k=1}^n \bigg [A_k C_{1}(x_k;\alpha _k)-B_k C_2(x_k;\beta _k)\bigg ],\\ C^*(\mathbf{x} ; {\varvec{\Theta }})& = \prod _{k=1}^n A_k C_1(x_k;\alpha _k)- \prod _{k=1}^n B_k C_2(x_k;\beta _k), \end{aligned}$$

are complex c.f.s in \({\mathbb {R}}^n,\) where \(\mathbf{A} =(A_1,\ldots ,A_n), \mathbf{B} =(B_1,\ldots ,B_n), \varvec{\alpha }=(\alpha _1,\ldots ,\alpha _n), \varvec{\beta }=(\beta _1,\ldots ,\beta _n)\) and \({\varvec{\Theta }}=(\mathbf{A} , \mathbf{B} , {\varvec{\alpha }, {\varvec{\beta }})}\) are vectors of parameters, \(\mathbf{x} =(x_1, \ldots , x_n)\in {\mathbb {R}}^n,\) and

$$\begin{aligned}&C_1(x_k;\alpha _k)={{\sin (\alpha _k x_k)}\over {x_k}} + i {2\over x_k} {\sin ^2{\bigg ({{\alpha _k x_k}\over 2}\bigg )}}, \\&C_2(x_k;\beta _k)={{\sin (\beta _k x_k)}\over {x_k}} + i {2\over x_k} {\sin ^2{\bigg ({{\beta _k x_k}\over 2}\bigg )}}. \end{aligned}$$

Example 4

Let’s consider, in \({\mathbb {R}}\), the following spectral density functions, with \(0<\alpha _k<\beta _k, \quad 0<B_k<A_k, \omega _k \in {\mathbb {R}}, k=1,\ldots ,n\)

$$\begin{aligned} f_1(\omega _k; \alpha _k)& = \left\{ \begin{array}{ll} 0 &{} \omega _k<0\\ \exp (-\alpha _k\omega _k) &{} \omega _k \ge 0 \\ \end{array} \right. \end{aligned}$$
(23)
$$\begin{aligned} f_2(\omega _k; \beta _k)& = \left\{ \begin{array}{ll} 0 &{} \omega _k<0\\ \exp (-\beta _k\omega _k) &{} \omega _k \ge 0 \\ \end{array} \right. \end{aligned}$$
(24)

which are not even functions and let \(C_1\) and \(C_2\) be the corresponding c.f.s of \(f_1\) and \(f_2\), respectively; since: \(A_k f_1({\omega _k; \alpha _k})- B_k f_2({\omega _k; \beta _k})\ge 0,\quad \omega _k \in {\mathbb {R}},\) then the following difference:

$$\begin{aligned}&A_k C_1({x_k}; \alpha _k)- B_k C_2({x_k}; \beta _k)\nonumber \\&\quad =\int _{{\mathbb {R}}} \exp (i{\omega _k} {x_k})( A_k f_1( \omega _k;\alpha _k)- B_k f_2({\omega _k; \beta _k})) d{\omega _k}\nonumber \\&\quad = {{A_k \alpha _k}\over {(x_k^2+\alpha _k^2)}} - {{B_k \beta _k}\over {(x_k^2+\beta _k^2)}}+ i \Bigg [{{A_k x_k}\over {(x_k^2+\alpha _k^2)}} - {{B_k x_k}\over {(x_k^2+\beta _k^2)}}\Bigg ], x_k \in {\mathbb {R}}, k=1,\ldots ,n, \end{aligned}$$
(25)

is a complex c.f..

According to Corollaries 4 and 5, the following functions:

$$\begin{aligned} C(\mathbf{x} ; \varvec{\Theta })& = \prod _{k=1}^n \bigg [A_k C_{1}(x_k;\alpha _k)-B_k C_2(x_k;\beta _k)\bigg ],\\ C^*(\mathbf{x} ; {\varvec{\Theta }})& = \prod _{k=1}^n A_k C_1(x_k;\alpha _k)- \prod _{k=1}^n B_k C_2(x_k;\beta _k), \end{aligned}$$

are complex c.f.s, both defined in \({\mathbb {R}}^n,\) where \(\mathbf{A} =(A_1,\ldots ,A_n), \mathbf{B} =(B_1,\ldots ,B_n), \varvec{\alpha }=(\alpha _1,\ldots ,\alpha _n), \varvec{\beta }=(\beta _1,\ldots ,\beta _n) \, \text {and} \,{\varvec{\Theta }}=(\mathbf{A} , \mathbf{B} , {\varvec{\alpha }, {\varvec{\beta }})}\) are vectors of parameters, \(\mathbf{x} =(x_1, \ldots , x_n) \in {\mathbb {R}}^n,\) and

$$\begin{aligned}&C_1(x_k;\alpha _k)={{\alpha _k}\over {(x_k^2+\alpha _k^2)}}+i {{x_k}\over {(x_k^2+\alpha _k^2)}}, \\&C_2(x_k;\beta _k)={{\beta _k}\over {(x_k^2+\beta _k^2)}}+i {{x_k}\over {(x_k^2+\beta _k^2)}}. \end{aligned}$$

Example 5

Let’s consider, in \({\mathbb {R}}\), the spectral density functions defined in the Eqs. (23) and (22) respectively, with \(\alpha _k>0, \beta _k>0, A_k>0, B_k>0, k=1,\ldots ,n\); note that \(f_1\) and \(f_2\) are not even functions and define:

$$\begin{aligned} f(\omega _k; \alpha _k, \beta _k,A_k,B_k)=A_k f_1(\omega _k; \alpha _k)-B_k f_2(\omega _k; \beta _k), \quad \omega _k \in {\mathbb {R}}, k=1,\ldots ,n; \end{aligned}$$
(26)

this last function is a spectral density function if \(0<B_k<A_k\) and

$$\begin{aligned} \beta _k= {\ln \big ({ A_k\over B_k}\big ) \over \alpha _k}, \quad k=1,\ldots ,n. \end{aligned}$$

If \(C_1\) and \(C_2\) are the corresponding c.f.s of \(f_1\) and \(f_2\), respectively, then the following difference:

$$\begin{aligned}&A_k C_1({x_k}; \alpha _k)- B_k C_2({x_k}; \beta _k)\nonumber \\&\quad =\int _{{\mathbb {R}}} \exp (i{\omega _k} {x_k})( A_k f_1( \omega _k;\alpha _k)- B_k f_2({\omega _k; \beta _k})) d{\omega _k}\nonumber \\&\quad = {{A_k \alpha _k}\over {(x_k^2+\alpha _k^2)}} - B_k {{\sin (\beta _k x_k)}\over {x_k}} + i \Bigg [{{A_k x_k}\over {(x_k^2+\alpha _k^2)}} - B_k {2\over x_k} {\sin ^2{\bigg ({{\beta _k x_k}\over 2}\bigg )}}\Bigg ], x_k\in {\mathbb {R}}, k=1,\ldots ,n \end{aligned}$$
(27)

with \(A_k C_1(0; \alpha _k)- B_k C_2(0; \beta _k)={{A_k }\over {\alpha _k}} - B_k \beta _k\), is a complex c.f..

According to Corollaries 4 and 5, the following functions:

$$\begin{aligned} C(\mathbf{x} ; \varvec{\Theta })& = \prod _{k=1}^n \bigg [A_k C_{1}(x_k;\alpha _k)-B_k C_2(x_k;\beta _k)\bigg ],\\ C^*(\mathbf{x} ; {\varvec{\Theta }})& = \prod _{k=1}^n A_k C_1(x_k;\alpha _k)- \prod _{k=1}^n B_k C_2(x_k;\beta _k), \end{aligned}$$

are complex c.f.s, both defined in \({\mathbb {R}}^n,\) where \(\mathbf{A} =(A_1,\ldots ,A_n), \mathbf{B} =(B_1,\ldots ,B_n), \varvec{\alpha }=(\alpha _1,\ldots ,\alpha _n), \varvec{\beta }=(\beta _1,\ldots ,\beta _n)\quad {\text {and}}\quad {\varvec{\Theta }}=(\mathbf{A} , \mathbf{B} , {\varvec{\alpha }, {\varvec{\beta }})}\) are vectors of parameters, \(\mathbf{x} =(x_1, \ldots , x_n)\in {\mathbb {R}}^n,\) and

$$\begin{aligned}&C_1(x_k;\alpha _k)= {{\alpha _k}\over {(x_k^2+\alpha _k^2)}}+i {{x_k}\over {(x_k^2+\alpha _k^2)}} , \\&C_2(x_k;\beta _k)={{\sin (\beta _k x_k)}\over {x_k}} + i {2\over x_k} {\sin ^2{\bigg ({{\beta _k x_k}\over 2}\bigg )}}. \end{aligned}$$

Example 6

Let’s consider, in \({\mathbb {R}}\), the spectral density functions

$$\begin{aligned} f_1(\omega _k)=\exp (-\alpha _k|\omega _k|), \quad \alpha _k>0, \end{aligned}$$

and \(f_2\) defined in Eq. (22), where \(f_1\) is an even function and \(f_2\) is not an even function and define, with \(\alpha _k>0, \beta _k>0, A_k>0, B_k>0, k=1,\ldots ,n\):

$$\begin{aligned}&f(\omega _k; \alpha _k, \beta _k,A_k,B_k)=A_k f_1(\omega _k; \alpha _k)-B_k f_2(\omega _k; \beta _k), \nonumber \\&\quad \omega _k \in {\mathbb {R}}, k=1,\ldots ,n; \end{aligned}$$
(28)

this last function is a spectral density function if \(0<B_k<A_k\) and \({\beta _k={\ln \left ( {A_k\over B_k}\right) \over \alpha _k}},\quad k=1,\ldots ,n.\) If \(C_1\) and \(C_2\) are the corresponding c.f.s of \(f_1\) and \(f_2\), respectively, then the following difference:

$$\begin{aligned}&A_k C_1({x_k}; \alpha _k)- B_k C_2({x_k}; \beta _k)\nonumber \\&\quad =\int _{{\mathbb {R}}} \exp (i{\omega _k} {x_k})( A_k f_1( \omega _k;\alpha _k)- B_k f_2({\omega _k; \beta _k})) d{\omega _k}\nonumber \\&\quad = {{2 A_k \alpha _k}\over {(x_k^2+\alpha _k^2)}} - B_k {{\sin (\beta _k x_k)}\over {x_k}} - i B_k {2\over x_k} {\sin ^2{\bigg ({{\beta _k x_k}\over 2}\bigg )}},\quad x_k \in {\mathbb {R}}, k=1,\ldots ,n, \end{aligned}$$
(29)

is a complex c.f.. According to Corollaries 4 and 5, the following functions:

$$\begin{aligned} C(\mathbf{x} ; \varvec{\Theta })& = \prod _{k=1}^n \bigg [A_k C_{1}(x_k;\alpha _k)-B_k C_2(x_k;\beta _k)\bigg ],\\ C^*(\mathbf{x} ; {\varvec{\Theta }})& = \prod _{k=1}^n A_k C_1(x_k;\alpha _k)- \prod _{k=1}^n B_k C_2(x_k;\beta _k), \end{aligned}$$

are complex c.f.s, where \(\mathbf{A} =(A_1,\ldots ,A_n), \mathbf{B} =(B_1,\ldots ,B_n), \varvec{\alpha }=(\alpha _1,\ldots ,\alpha _n), \varvec{\beta }=(\beta _1,\ldots ,\beta _n)\quad {\text {and}} {\varvec{\Theta }}=(\mathbf{A} , \mathbf{B} , {\varvec{\alpha }, {\varvec{\beta }})}\) are vectors of parameters, \(\mathbf{x} =(x_1, \ldots , x_n)\in {\mathbb {R}}^n,\) and \(C_1(x_k;\alpha _k)= {{2 \alpha _k}\over {(x_k^2+\alpha _k^2)}}, \quad C_2(x_k;\beta _k)= {{\sin (\beta _k x_k)}\over {x_k}} + i {2\over x_k} {\sin ^2{\bigg ({{\beta _k x_k}\over 2}\bigg )}}.\)

3.2 Difference of two real covariance functions

In the following, some applications of Corollary 3, 4 and 5 are given for the subset of the real c.f.s: hence, \(C_1\) and \(C_2\) defined in the same corollaries are real c.f.s which depend on some parameters. In particular, starting from some well known c.f.s (exponential, Gaussian and rational models), the difference of the various combinations of these last models has been analyzed: it will be shown that the difference of two selected c.f.s could be a c.f. for some values of the parameters on which these models depend. On the other hand, the same difference could not be a c.f. for other values of these parameters. The models obtained as the difference of two c.f.s could be negative in a subset of the domain on which the c.f.s are defined. The proposed models \(C_i, i=1, 2,\) are integrable, hence, according to (3), the spectral density functions can be expressed as the Fourier transform of these covariance models.

Example 7

Let

$$\begin{aligned} C_1({x_k};\alpha _k)& = \exp \big (-\alpha _k x_k^2\big ), \quad x_k \in {\mathbb {R}}, \quad \alpha _k>0, k=1,\ldots ,n,\\ C_2({x_k}; \beta _k)& = \exp \big (-\beta _k x_k^2\big ), \quad x_k \in {\mathbb {R}}\quad \beta _k>0, k=1,\ldots ,n, \end{aligned}$$

be Gaussian c.f.s; these models are integrable, then their Fourier transforms \(f_1\) and \(f_2\) are, respectively:

$$\begin{aligned} f_1({\omega _k};\alpha _k)& = {1\over {2\pi }}\int _{{\mathbb {R}}}\exp \big (-i x_k \omega _k\big ) \exp \big (-\alpha _k x_k^2\big ) dx_k \\& = {{1\over {2}\sqrt{\pi \alpha _k}}} \exp \bigg [-\bigg ({\omega _k^2\over {4\alpha _k}} \bigg )\bigg ],\\ f_2({\omega _k};\beta _k)& = {1\over {2\pi }}\int _{{\mathbb {R}}}\exp \big (-i x_k \omega _k\big ) \exp \big (-\beta _k x_k^2\big ) dx_k \\& = {{1\over {2}\sqrt{\pi \beta _k}}} \exp \bigg [-\bigg ({\omega _k^2\over {4\beta _k}} \bigg )\bigg ], \end{aligned}$$

\(\omega _k \in {\mathbb {R}}, \quad k=1,\ldots ,n.\) Then

$$\begin{aligned} C({x_k};A_k, B_k, \alpha _k,\beta _k) = A_k C_1({x_k}; \alpha _k)- B_k C_2({x_k}; \beta _k), \quad x_k \in {\mathbb {R}}, A_k>0, B_k>0, \end{aligned}$$
(30)

according to Corollaries 3 and 4, is a c.f., for each \(k=1,\ldots ,n\), if its Fourier transform is a spectral density function, i.e., \(A_k f_1(\omega _k; \alpha _k)- B_k f_2(\omega _k; \beta _k)\ge 0,\quad \omega _k \in {\mathbb {R}};\) the previous inequality is satisfied if:

$$\begin{aligned} {\omega _k^2\over {4}}\bigg ({1\over {\beta _k}}-{1\over {\alpha _k}}\bigg ) > \ln \bigg ({B_k\over {A_k}} \sqrt{{\alpha _k \over {\beta _k}}}\bigg ); \end{aligned}$$
(31)

then, (31) is always satisfied if \(\alpha _k >\beta _k\quad\) and \(\quad {{B_k\over {A_k}} <\sqrt{{\beta _k\over {\alpha _k}}}}, k=1,\ldots ,n.\)

Example 8

Given the following exponential c.f.s:

$$\begin{aligned} C_1({x_k}; \alpha _k)& = \exp \big (-\alpha _k |x_k| \big ),\quad x_k \in {\mathbb {R}}, \alpha _k>0, k=1,\ldots ,n, \\ C_2({x_k}; \beta _k)& = \exp \big (-\beta _k |x_k| \big ), \quad x_k \in {\mathbb {R}}, \beta _k>0, k=1,\ldots ,n; \end{aligned}$$

then, \(C_1\) and \(C_2\) are integrable and

$$\begin{aligned}&C({x_k};A_k, B_k, \alpha _k,\beta _k)\\=A_k \exp \big (-\alpha _k |x_k| \big ) \nonumber \; - B_k \exp \big (-\beta _k|x_k|\big ), \quad A_k>0, B_k>0, \end{aligned}$$
(32)

is a c.f. if its Fourier transform f is a spectral density function, i.e.,

$$\begin{aligned} f(\omega _k; A_k, B_k, \alpha _k,\beta _k)={A_k\over \pi }{ {\alpha _k\over {(\omega _k^2+\alpha _k^2)}}}-{ B_k\over \pi }{ {\beta _k\over {(\omega _k^2+\beta _k^2)}}}\ge 0, \quad \omega _k \in {\mathbb {R}}; \end{aligned}$$
(33)

in particular, (33) is satisfied if:

$${ {(\omega _k^2+\beta _k^2)\over {(\omega _k^2+\alpha _k^2)}}}-{ B_k\over A_k}{ {\beta _k\over {\alpha _k}}}\ge 0, \quad \omega _k \in {\mathbb {R}}.$$

According to the values of the parameters \(\alpha _k\) and \(\beta _k, k=1,\ldots ,n\), some special situations can be analyzed:

  1. 1.

    \(\alpha _k>\beta _k, k=1,\ldots ,n\); in this case, it is easy to verify that the minimum value of the function:

    $$\zeta (\omega _k; \alpha _k,\beta _k)={ {(\omega _k^2+\beta _k^2)\over {(\omega _k^2+\alpha _k^2)}}}$$

    is obtained for \(\omega _k^*=0\); hence \(\zeta (0; \alpha _k, \beta _k)={ {\beta _k^2\over {\alpha _k^2}}};\) as a consequence, (33) is a spectral density function if: \(\quad {B_k\over A_k }<{ {\beta _k\over {\alpha _k}}}<1.\)

  2. 2.

    \(\alpha _k<\beta _k, k=1,\ldots ,n\); in this case, \(\zeta (\omega _k; \alpha _k, \beta _k)={ {(\omega _k^2+\beta _k^2)\over {(\omega _k^2+\alpha _k^2)}}}>1, \quad \omega _k\in {\mathbb {R}},\) then (33) is a spectral density function if:

    $$\begin{aligned} {{B_k\over A_k }<{ {\alpha _k\over {\beta _k}}}}. \end{aligned}$$
    (34)

Example 9

Given the following c.f.s,

$$\begin{aligned} C_1({x_k};\alpha _k)= {1 \over {(x_k^2+\alpha _k^2)}}, \quad C_2({x_k}; \beta _k)= {1 \over {(x_k^2+\beta _k^2)}}, \quad x_k \in {\mathbb {R}}, \alpha _k>0, \beta _k>0, \end{aligned}$$

then \(C_1\) and \(C_2\) are integrable and

$$\begin{aligned} C(x_k;A_k, B_k, \alpha _k,\beta _k)= {A_k \over {(x_k^2+\alpha _k^2)}}- {B_k \over {(x_k^2+\beta _k^2)}}, \quad A_k>0, B_k>0, \end{aligned}$$
(35)

is a c.f. if its Fourier transform f is a spectral density function, i.e.,

$$\begin{aligned} f(\omega _k; A_k,B_k,\alpha _k, \beta _k)\\={A_k\over {2\alpha _k}} \exp \big (-\alpha _k |\omega _k|\big )-{B_k\over {2\beta _k}} \exp \big (-\beta _k|\omega _k|\big )\ge 0, \quad \omega _k \in {\mathbb {R}}; \end{aligned}$$
(36)

in particular, (36) is satisfied if \(\beta _k>\alpha _k,\) and \({B_k\over A_k}<{ {\beta _k\over {\alpha _k}}}, k=1,\ldots ,n.\)

Example 10

Given the following c.f.s:

$$\begin{aligned} C_1({x_k}; \alpha _k)= \exp \big (-\alpha _k|x_k|\big ),\quad C_2({x_k};\alpha _k)={{\pi }\over {2 \alpha _k^3}} (1+\alpha _k|x_k|)\big [\exp \big (-\alpha _k |x_k|\big )\big ], \end{aligned}$$

with \(x_k \in {\mathbb {R}}, \alpha _k>0, k=1,\ldots ,n;\) then, \(C_1\) and \(C_2\) are integrable and

$$\begin{aligned} C(x_k; A_k, B_k, \alpha _k)& = A_k C_1({x_k}; \alpha _k)-B_k C_2({x_k};\alpha _k)\nonumber \\& = \exp \big (- \alpha _k|x_k|\big )\bigg [ A_k - {{ \pi B_k }\over {2\alpha _k^3}} (1+\alpha _k|x_k|)\bigg ], \quad A_k>0, B_k>0, \end{aligned}$$
(37)

is a c.f. if its Fourier transform f is a spectral density function, i.e.,

$$\begin{aligned} f(\omega _k;A_k, B_k, \alpha _k)= {A_k\over \pi }{ {\alpha _k\over {(\omega _k^2+\alpha _k^2)}}\Bigg [1-{{\pi B_k}\over {A_k \alpha _k(\omega _k^2+\alpha _k^2)}}}\Bigg ]\ge 0, \quad \omega _k \in {\mathbb {R}}. \end{aligned}$$

Note that the maximum value of the function

$$\epsilon (\omega _k; \alpha _k)={1\over {(\omega _k^2+\alpha _k^2)}}$$

is obtained for \(\omega _k=0\), then f is a spectral density function, hence C is a c.f. if:

$$\quad {B_k \over A_k}<{{ \alpha _k^3}\over {\pi }}.$$

Note that:

$$\quad C^*({x_k}; A_k,B_k, \alpha _k)=B_k C_2({x_k}; \alpha _k)-A_k C_1({x_k}; \alpha _k), \quad A_k>0, B_k>0,$$

cannot be a c.f. because its Fourier transform:

$$\begin{aligned} f(\omega _k;A_k,B_k, \alpha _k)\\= {A_k\over \pi }{ {\alpha _k\over {(\omega _k^2+\alpha _k^2)}}\Bigg [{{\pi B_k}\over {A_k \alpha _k(\omega _k^2+\alpha _k^2)}}}-1\Bigg ], \quad \omega _k \in {\mathbb {R}}, \end{aligned}$$

cannot be a spectral density function, since the function: \(\epsilon (\omega _k; \alpha _k)={1\over {(\omega _k^2+\alpha _k^2)}},\) does not present a minimum value.

Example 11

Given the following c.f.s

$$\begin{aligned} C_1({x_k}; \alpha _k)= \exp \big (- \alpha _k |x_k|\big ), \quad C_2({x_k};\beta _k)= \exp \big (- \beta _k x_k^2\big ), \quad x_k \in {\mathbb {R}}, \alpha _k>0, \beta _k>0, \end{aligned}$$

\(k=1,\ldots ,n,\) then \(C_1\) and \(C_2\) are integrable and

$$\begin{aligned} C({x_k}; A_k,B_k,\alpha _k, \beta _k)=A_k \exp \big (- \alpha _k |x_k|\big )-B_k \exp \big (- \beta _k x_k^2\big ), \quad A_k>0, B_k>0, \end{aligned}$$
(38)

is a c.f., for each \(k=1,\ldots ,n\), if the corresponding Fourier transform f is a spectral density function, i.e.,

$$\begin{aligned} f(\omega _k; A_k,B_k,\alpha _k, \beta _k)\\= \bigg [{A_k\over \pi } {\alpha _k \exp (\omega _k^2/{4 \beta _k})\over {(\omega _k^2+\alpha _k^2)}} - {B_k\over {2\sqrt{\pi \beta _k}}}\bigg ] {\exp \bigg (- {\omega _k^2\over {4 \beta _k}}\bigg )}\ge 0, \quad \omega _k \in {\mathbb {R}}. \end{aligned}$$

According to the values of the parameters \(\alpha _k\) and \(\beta _k, k=1,\ldots ,n\), some special situations can be analyzed:

  1. 1.

    \(\alpha _k^2 -4\beta _k>0, k=1,\ldots ,n\); in this case, it is easy to verify that the minimum value of the function: \(\kappa (\omega _k, \alpha _k, \beta _k)= {\alpha _k \exp (\omega _k^2/{4 \beta _k})\over {(\omega _k^2+\alpha _k^2)}}\) is obtained for \(\omega _k^*=0\); then, taking into account the inequalities \(\alpha _k^2 -4\beta _k>0, k=1,\ldots ,n\), f is a spectral density function, as a consequence C defined in (38) is a c.f., if

    $$\begin{aligned} 0<{B_k\over A_k}<{2 \sqrt{\pi }\over \pi } {\sqrt{\beta _k} \over {\alpha _k }}<{1\over \sqrt{\pi }}, \quad hence: 0<{B_k\over A_k}<{1\over \sqrt{\pi }}\approx 0.564. \end{aligned}$$
  2. 2.

    \(\alpha _k^2 -4\beta _k<0, k=1,\ldots ,n\); in this case, the minimum value of the function: \(\kappa ( \omega _k, \alpha _k,\beta _k)= {\alpha _k \exp (\omega _k^2/{4 \beta _k})\over {(\omega _k^2+\alpha _k^2)}}\) is obtained for \(\omega _k^*=\sqrt{4\beta _k-\alpha _k^2};\) then, taking into account the inequalities \(\alpha _k^2 -4\beta _k<0, k=1,\ldots ,n\), f is a spectral density function, as a consequence C defined in (38) is a c.f., if \(0<{B_k\over A_k}<{e \over \sqrt{\pi }}\approx 1.53.\)

Note that:

$$C({x_k};A_k,B_k,\alpha _k, \beta _k)=B_k C_2({x_k}; \beta _k)-A_k C_1({x_k}; \alpha _k), \quad A_k>0, B_k>0,$$

cannot be a c.f. because its Fourier transform:

$$\begin{aligned} f(\omega _k;A_k,B_k,\alpha _k, \beta _k)= \bigg [{B_k\over {2\sqrt{\pi \beta _k }}}-{A_k\over \pi } {\alpha _k \exp (\omega _k^2/{4 \beta _k})\over {(\omega _k^2+\alpha _k^2)}} \bigg ] {\exp \bigg (- {\omega _k^2\over {4 \beta _k}}\bigg )}, \quad \omega _k \in {\mathbb {R}}, \end{aligned}$$

cannot be a spectral density, since the function: \(\kappa ( \omega _k; \alpha _k, \beta _k)= {\alpha _k \exp (\omega _k^2/{4 \beta _k})\over {(\omega _k^2+\alpha _k^2)}}\) does not present an absolute maximum value.

Example 12

Given the c.f.s

$$\begin{aligned} C_1({x_k}; \alpha _k)=\exp \big (- \alpha _k |x_k| \big ),\quad C_2({x_k}; \beta _k)= {1 \over {(x_k^2+\beta _k^2)}}, \quad x_k \in {\mathbb {R}}, \alpha _k>0, \beta _k>0, \end{aligned}$$

\(k=1,\ldots ,n\); then \(C_1\) and \(C_2\) are integrable and

$$\begin{aligned} C({x_k}; A_k,B_k,\alpha _k, \beta _k)=A_k \exp \big (- \alpha _k |x_k| \big )- {B_k\over {(x_k^2+\beta _k^2)}}, \quad A_k>0, B_k>0, \end{aligned}$$
(39)

is a c.f. if the corresponding Fourier transform f is a spectral density function, i.e.,

$$\begin{aligned} f(\omega _k;A_k,B_k,\alpha _k, \beta _k)\\=\Bigg [{A_k \over \pi } {\alpha _k \exp (\beta _k|\omega _k|)\over { (\omega _k^2+\alpha _k^2)}} - {B_k\over {2 \beta _k}}\Bigg ] {1\over {\exp \big [ (\beta _k|\omega _k|)\big ]}}\ge 0, \quad \omega _k \in {\mathbb {R}}. \end{aligned}$$

According to the values of the parameters \(\alpha _k\) and \(\beta _k, k=1,\ldots ,n\), some special situations can be analyzed:

  1. 1.

    \(\alpha _k \beta _k>1, k=1,\ldots ,n\); in this case, the minimum value of the function

    $$\begin{aligned} \chi ( \omega _k;\alpha _k, \beta _k)= {\alpha _k \exp (\beta _k|\omega _k|)\over { (\omega _k^2+\alpha _k^2)}}, \end{aligned}$$
    (40)

    is obtained for \(\omega _k^*=0\); hence, f is a spectral density function, as a consequence C defined in (39) is a c.f. if: \(\quad 0<{B_k\over A_k}< {2 \over \pi } {\beta _k \over \alpha _k}<{2 \over \pi } \beta _k^2.\)

  2. 2.

    \(\alpha _k \beta _k<1, k=1,\ldots ,n\); in this case, there exist two minimum values for the function (40): \(\omega _k^*=0\) and \(\omega _k^{**}={{1+\sqrt{1-\alpha _k^2 \beta _k^2}}\over {\beta _k}}.\) By comparing the values of the function \(\chi\), defined in (40), computed in \(\omega _k^*\) and \(\omega _k^{**}\), i.e.,

    $$\begin{aligned} \chi ( \omega _k^{**};\alpha _k, \beta _k)-\chi ( \omega _k^{*}; \alpha _k, \beta _k)>0, \end{aligned}$$
    (41)

    it results that (41) is satisfied if:

    $$\begin{aligned} (\alpha _k \beta _k)^2 \exp \bigg (1+\sqrt{1-(\alpha _k \beta _k)^2}\bigg )>2\bigg (1+\sqrt{1-(\alpha _k \beta _k)^2}\bigg ), \quad k=1,\ldots ,n. \end{aligned}$$
    (42)

    Inequality (42) is satisfied for \(0.8<\alpha _k \beta _k<1\), whereas for \(0<\alpha _k \beta _k<0.8\) it results \(\chi ( \omega _k^{**}; \alpha _k, \beta _k)-\chi ( \omega _k^{*}; \alpha _k, \beta _k)<0.\) Hence, according to the previous results:

    • \(0.8<\alpha _k \beta _k<1\), expression (39) is a c.f. if: \(\quad 0<{B_k\over A_k}< {2\over \pi } {\beta _k \over \alpha _k}<{2 \over \pi } {1\over \alpha _k^2};\)

    • \(0<\alpha _k \beta _k<0.8\), expression (39) is a c.f. if: \(\quad 0<{B_k\over A_k}< {2 \over \pi } \chi ( \omega _k^{**};\alpha _k, \beta _k) {\beta _k }.\)

Note that: \(C({x};A_k,B_k,\alpha _k, \beta _k)=B_kC_2({x_k}; \beta _k)-A_k C_1({x_k}; \alpha _k), \quad A_k>0, B_k>0,\) cannot be a c.f. because its Fourier transform:

$$\begin{aligned} f( \omega _k;A_k,B_k,\alpha _k, \beta _k)\\=\Bigg [{B_k\over {2 \beta _k}}-{A_k \over \pi } {\alpha _k \exp (\beta _k|\omega _k|)\over { (\omega _k^2+\alpha _k^2)}}\Bigg ] {1\over {\exp (\beta _k|\omega _k|)}}, \quad \omega _k \in {\mathbb {R}}, \end{aligned}$$

cannot be a spectral density function, since the function

$$\chi (\omega _k; \alpha _k, \beta _k)= {\alpha _k \exp (\beta _k|\omega _k|)\over { (\omega _k^2+\alpha _k^2)}},$$

does not present an absolute maximum value.

Example 13

Given the following c.f.s,

$$\begin{aligned} C_1({x_k};\alpha _k)= {1 \over { (x_k^2+\alpha _k^2)}}, \quad C_2({x_k}; \beta _k)= \exp \big (- \beta _k x_k^2\big ), \quad x_k \in {\mathbb {R}}, \alpha _k>0, \beta _k>0, \end{aligned}$$

\(k=1,\ldots ,n\), then \(C_1\) and \(C_2\) are integrable and

$$\begin{aligned} C({x_k};A_k,B_k,\alpha _k, \beta _k)= {A_k \over { (x_k^2+\alpha _k^2)}}-B_k \exp \big (- \beta _k x_k^2\big ), \quad A_k>0, B_k>0, \end{aligned}$$
(43)

is a c.f. if its Fourier transform f is a spectral density function, i.e.,

$$\begin{aligned} f(\omega _k;A_k,B_k,\alpha _k, \beta _k)\\={A_k\over {2\alpha _k}} \exp \big (- \alpha _k |\omega _k|\big )-{B_k\over {2\sqrt{\pi \beta _k}}} \exp \bigg [-\bigg ( {\omega _k^2\over {4\beta _k}} \bigg )\bigg ]\ge 0, \omega _k \in {\mathbb {R}}, \end{aligned}$$
(44)

which is equivalent to the following inequality:

$$\begin{aligned} \exp \bigg ({\omega _k^2\over {4\beta _k}}-\alpha _k |\omega _k|\bigg )-{{B_k\alpha _k}\over {A_k\sqrt{\pi \beta _k}}} \ge 0, \quad \omega _k \in {\mathbb {R}}; \end{aligned}$$
(45)

the minimum value of the function \(\tau(\omega_k;\alpha_k,\beta_k)= \exp \bigg ({\omega _k^2\over {4\beta _k}}-\alpha _k |\omega _k|\bigg )\) is obtained for \(\omega _k^*=2 \alpha _k \beta _k\); hence, (44) is a spectral density function if: \(\exp (\alpha _k^2 \beta _k)<{{A_k\sqrt{\pi \beta _k}}\over {B_k\alpha _k}},\quad k=1,\ldots ,n.\)

Note that: \(C({x_k};A_k,B_k,\alpha _k, \beta _k)=B_k C_2({x_k}; \beta _k)-A_k C_1({x_k}; \alpha _k), \quad A_k>0, B_k>0,\) cannot be a c.f. because its Fourier transform:

$$\begin{aligned} f(\omega _k;A_k,B_k,\alpha _k, \beta _k)\\= {B_k\over {2\sqrt{\pi \beta _k}}} \exp \bigg [-\bigg ( {\omega _k^2\over {4\beta _k}} \bigg )\bigg ]-{A_k\over {2\alpha _k}} exp\big (- \alpha _k |\omega _k|\big ), \omega _k \in {\mathbb {R}}, \end{aligned}$$

cannot be a spectral density function, since the following inequality

$$\begin{aligned} {{B_k\alpha _k}\over {A_k\sqrt{\pi \beta _k}}}- \exp \bigg ({\omega _k^2\over {4\beta _k}}-\alpha _k |\omega _k|\bigg ) \ge 0, \quad \omega _k \in {\mathbb {R}}, \end{aligned}$$

is not satisfied, because the function \(\tau\) does not present an absolute maximum value.

In the same way as all that proposed in all the examples shown in the Sect. 3.1 and according to Corollaries 4 and 5, the following functions:

$$\begin{aligned} C(\mathbf{x} ; \varvec{\Theta })& = \prod _{k=1}^n \bigg [A_k C_{1}(x_k;\alpha _k)-B_k C_2(x_k;\beta _k)\bigg ],\\ C^*(\mathbf{x} ; {\varvec{\Theta }})& = \prod _{k=1}^n A_k C_1(x_k;\alpha _k)- \prod _{k=1}^n B_k C_2(x_k;\beta _k), \end{aligned}$$

are real c.f.s, both defined in \({\mathbb {R}}^n,\) where \(\mathbf{A} =(A_1,\ldots ,A_n), \mathbf{B} =(B_1,\ldots ,B_n), \varvec{\alpha }=(\alpha _1,\ldots ,\alpha _n), \varvec{\beta }=(\beta _1,\ldots ,\beta _n)\) and \({\varvec{\Theta }}=(\mathbf{A} , \mathbf{B} , {\varvec{\alpha }, {\varvec{\beta }})}\) are vectors of parameters, \(\mathbf{x} =(x_1, \ldots , x_n)\in {\mathbb {R}}^n,\) where \(C_1\) and \(C_2\) are the c.f.s defined in the Examples 7, 8, 9, 10, 11, 12 and 13.

3.3 Difference of isotropic covariance functions

As already underlined, an isotropic c.f. is necessarily a real valued c.f.. Although there exist several examples of real c.f.s for stationary random fields in one dimensional space, however some of the same c.f.s cannot be examples of isotropic c.f.s in any dimensional space (see for example, the triangular c.f.s). Nevertheless, there exist functions which can be isotropic c.f.s in any dimensional space \({\mathbb {R}}^n, n \in \mathbb {N}\). A linear combination, with negative weights, of two continuous spatial isotropic or spatio-temporal c.f.s has been analyzed by Ma (2005) in all dimensions: however, the two spatial or spatio-temporal c.f.s involved in the analysis belong to the same family. In this subsection some examples for the difference of two isotropic c.f.s are provided for the two and three dimensional spaces; in particular, for some of these examples the c.f.s involved in the difference could not belong to the same family.

The following equations express the multidimensional spectral density f in terms of the one-dimensional spectral density \(f_1\), for the two and three dimensional spaces, i.e., \(n=2\) and \(n=3\), which are the most relevant cases in the applications:

$$\begin{aligned} f(\omega )& = -{1\over \pi }\int _ {\omega }^\infty {df_1(\omega _1)\over {d\omega _1}}{ d\omega _1\over {(\omega _1^2-\omega ^2)^{1/2}}}, \quad n=2; \end{aligned}$$
(46)
$$\begin{aligned} f(\omega )& = -{1\over {2\pi \omega }}{df_1(\omega )\over {d\omega }}, \quad n=3. \end{aligned}$$
(47)

To check whether or not a given function C is an n-dimensional isotropic c.f. (for \(n=2\) and \(n=3\)) it is important to find, first of all, the one dimensional Fourier transform \(f_1\) of the function C, then computing the corresponding two and three dimensional spectral density f from \(f_1\), utilizing Eqs. (46) and (47) and finally verifying whether or not the function f is everywhere non negative.

The exponential and Gaussian c.f.s are isotropic c.f.s in the space \({\mathbb {R}}^n\), for any \(n\in \mathbb {N}\). In the following examples, starting from the one dimensional spectral density functions given in the Examples 7 and 8 and applying Eqs. (46) and (47), the difference, considering various combinations, between the isotropic exponential and Gaussian c.f.s have been analyzed for the two and three dimensional spaces.

Example 14

In the following example, the difference between two isotropic Gaussian c.f.s has been considered in the Euclidean spaces \({\mathbb {R}}^2\) and \({\mathbb {R}}^3\), respectively.

  • \(n=2\). In the Example 7, given the one dimensional c.f.s \(C_1(x_k)\) and \(C_2(x_k)\) and the corresponding one dimensional spectral density functions \(f_1(\omega _k)\) and \(f_2(\omega _k)\), by utilizing Eq. (46), the two dimensional spectral density functions \(f_1(\omega )\) and \(f_2(\omega )\) can be obtained:

    $$\begin{aligned} f_1(\omega ;\alpha )= {1\over {4\pi \alpha }} \exp \bigg [-\bigg ({\omega ^2\over {4\alpha }} \bigg )\bigg ],\quad f_2(\omega ;\beta )= {1\over {4\pi \beta }} \exp \bigg [-\bigg ({\omega ^2\over {4\beta }} \bigg )\bigg ], \end{aligned}$$
    (48)

    where \(\alpha>0,\quad \beta >0, \quad \varvec{\omega }=(\omega _1,\omega _2)\) and \(\omega =\sqrt{\omega _1^2+\omega _2^2}.\) Then

    $$\begin{aligned} C({x};A, B, \alpha ,\beta ) = A \exp \big (-\alpha x^2\big )- B\exp \big (-\beta x^2\big ), A>0, B>0, \end{aligned}$$
    (49)

    according to Corollary 3, is a c.f. if its Fourier transform is a spectral density function, i.e.,

    $$\begin{aligned} A f_1(\omega ; \alpha )- Bf_2(\omega ;\beta )\\={A\over {4\pi \alpha }} \exp \bigg [-\bigg ({\omega ^2\over {4\alpha }} \bigg )\bigg ]- {B\over {4\pi \beta }} \exp \bigg [-\bigg ({\omega ^2\over {4\beta }} \bigg )\bigg ]\ge 0,\quad \omega \in {\mathbb {R}}; \end{aligned}$$

    the previous inequality is satisfied if:

    $$\begin{aligned} {\omega ^2\over {4}}\bigg ({1\over {\beta }}-{1\over {\alpha }}\bigg ) > \ln \bigg ({B\over {A}} {\alpha \over {\beta }}\bigg ); \end{aligned}$$
    (50)

    then, (50) is always satisfied if \(\alpha >\beta\) and \({ B\over {A}} {\alpha \over {\beta }}<1,\) i.e., \(1< {\alpha \over {\beta }}<{ A\over {B}}.\)

  • \(n=3\). In this case, utilizing Eq. (47), the following three dimensional spectral density functions \(f_1(\omega )\) and \(f_2(\omega )\) can be obtained:

    $$\begin{aligned} f_1(\omega ;\alpha )= {1\over {8({\pi \alpha })^{2/3}}} \exp \bigg [-\bigg ({\omega ^2\over {4\alpha }} \bigg )\bigg ]; \\ f_2(\omega ;\beta )= {1\over {8({\pi \beta })^{2/3}}} \exp \bigg [-\bigg ({\omega ^2\over {4\beta }} \bigg )\bigg ], \end{aligned}$$
    (51)

    where \(\varvec{\omega }=(\omega _1,\omega _2,\omega _3), \omega =\sqrt{\omega _1^2+\omega _2^2+\omega _3^2}, \alpha>0, \beta >0.\) Then

    $$\begin{aligned} C({x};A, B, \alpha ,\beta ) \\= A \exp \big (-\alpha x^2\big )- B\exp \big (-\beta x^2\big ), A>0, B>0, \end{aligned}$$
    (52)

    according to Corollary 3, is a c.f. if its Fourier transform is a spectral density function, i.e.,

    $$\begin{aligned}&A f_1(\omega ; \alpha )- Bf_2(\omega \beta )\\&\quad ={A\over {8({\pi \alpha })^{2/3}}} \exp \bigg [-\bigg ({\omega ^2\over {4\alpha }} \bigg )\bigg ]- {B\over {8({\pi \beta })^{2/3}}} \exp \bigg [-\bigg ({\omega ^2\over {4\beta }} \bigg )\bigg ]\ge 0,\quad \omega \in {\mathbb {R}}; \end{aligned}$$

    the previous inequality is satisfied if:

    $$1< {\alpha \over {\beta }}<{ \left ({A\over {B}}\right )^{3/2}}.$$

Example 15

In the following example, the difference between two isotropic exponential c.f.s has been considered in the Euclidean spaces \({\mathbb {R}}^2\) and \({\mathbb {R}}^3,\) respectively.

  • \(n=2\). In the Example 8, given the c.f.s \(C_1(x_k)\) and \(C_2(x_k)\) and the corresponding one dimensional spectral density functions \(f_1(\omega _k)\) and \(f_2(\omega _k)\), by utilizing Eq. (46), the two dimensional spectral density functions \(f_1(\omega )\) and \(f_2(\omega )\) can be obtained: \(f_1(\omega ;\alpha )={\alpha \over {2\pi (\omega ^2+\alpha ^2)^{3/2}}}, \quad f_2(\omega ;\beta ) ={\beta \over {2\pi (\omega ^2+\beta ^2)^{3/2}}},\) where \(\alpha>0,\quad \beta >0, \quad \varvec{\omega }=(\omega _1,\omega _2)\) and \(\omega =\sqrt{\omega _1^2+\omega _2^2}.\) Then

    $$\begin{aligned} C({x};A, B, \alpha ,\beta )=A \exp \big (-\alpha |x| \big ) - B \exp \big (-\beta |x|\big ), \quad A>0, B>0, \end{aligned}$$
    (53)

    according to Corollary 3, is a c.f. if its Fourier transform is a spectral density function, i.e.,

    $$\begin{aligned} {A\alpha \over {2\pi (\omega ^2+\alpha ^2)^{3/2}}}-{B\beta \over {2\pi (\omega ^2+\beta ^2)^{3/2}}}\ge 0, \quad \omega \in {\mathbb {R}}; \end{aligned}$$
    (54)

    in particular, (54) is satisfied if: \(\bigg ({{\omega ^2+\beta ^2}\over {\omega ^2+\alpha ^2}}\bigg )^{3/2}-{ B\over A}{ {\beta \over {\alpha }}}\ge 0, \quad \omega \in {\mathbb {R}}.\)

    If \(\alpha <\beta\), then (54) is satisfied if: \(\quad 1< {\beta \over \alpha } < {A\over B}\); on the other hand, if \(\alpha >\beta\), then (54) is satisfied if: \(1< {\alpha \over \beta } < \sqrt{A\over B}.\)

  • \(n=3.\) In this case, utilizing Eq. (47), the following three dimensional spectral density functions \(f_1\) and \(f_2\) can be obtained:

    $$\begin{aligned} f_1(\omega ;\alpha )={\alpha \over {\pi ^2(\omega ^2+\alpha ^2)^{2}}},\quad f_2(\omega ;\beta ) ={\beta \over {\pi ^2(\omega ^2+\beta ^2)^{2}}}, \end{aligned}$$

    where \(\varvec{\omega }=(\omega _1,\omega _2,\omega _3), \omega =\sqrt{\omega _1^2+\omega _2^2+\omega _3^2}, \alpha>0, \beta >0.\) Then

    $$\begin{aligned} C({x};A, B, \alpha ,\beta )=A \exp \big (-\alpha |x| \big ) - B \exp \big (-\beta |x|\big ), \quad A>0, B>0, \end{aligned}$$
    (55)

    according to Corollary 3, is a c.f. if its Fourier transform is a spectral density function, i.e.,

    $$\begin{aligned} {A\alpha \over {\pi ^2(\omega ^2+\alpha ^2)^{2}}}-{B\beta \over {\pi ^2(\omega ^2+\beta ^2)^{2}}}\ge 0, \quad \omega \in {\mathbb {R}}; \end{aligned}$$
    (56)

    in particular, (56) is satisfied if: \(\bigg ({{\omega ^2+\beta ^2}\over {\omega ^2+\alpha ^2}}\bigg )^{2}-{ B\over A}{ {\beta \over {\alpha }}}\ge 0, \quad \omega \in {\mathbb {R}}.\)

    If \(\alpha <\beta\), then (56) is satisfied if: \(\quad 1< {\beta \over \alpha } < {A\over B}\); on the other hand, if \(\alpha >\beta\), then (56) is satisfied if: \(1< {\alpha \over \beta } <\big ( {A\over B}\big )^{1/3}.\)

Example 16

In the following example, the difference between an isotropic exponential and an isotropic Gaussian c.f.s has been considered in \({\mathbb {R}}^2\) and \({\mathbb {R}}^3\), respectively.

  • \(n=2\). In the Examples 8 and 7, given the one dimensional c.f.s \(C_1(x_k)\) and \(C_2(x_k)\) for the exponential and the Gaussian models, respectively, with the corresponding one dimensional spectral density functions \(f_1(\omega _k)\) and \(f_2(\omega _k)\), by utilizing Eq. (46), the two dimensional spectral density functions \(f_1(\omega )\) and \(f_2(\omega )\), with \(\alpha =\beta\), can be obtained:

    $$\begin{aligned} f_1(\omega ;\alpha )={\alpha \over {2\pi (\omega ^2+\alpha ^2)^{3/2}}}, \quad f_2(\omega ;\alpha )= {1\over {4\pi \alpha }} \exp \bigg [-\bigg ({\omega ^2\over {4\alpha }} \bigg )\bigg ], \alpha >0, \end{aligned}$$

    where \(\alpha >0, \quad \varvec{\omega }=(\omega _1,\omega _2)\) and \(\omega =\sqrt{\omega _1^2+\omega _2^2}.\) Then

    $$\begin{aligned} C({x};A, B, \alpha ) = A \exp \big (-\alpha |x|\big )- B\exp \big (-\alpha x^2\big ), A>0, B>0, \end{aligned}$$
    (57)

    according to Corollary 3, is a c.f. if its Fourier transform is a spectral density function, i.e., \({A\alpha \over {2\pi (\omega ^2+\alpha ^2)^{3/2}}}- {B\over {4\pi \alpha }} \exp \bigg [-\bigg ({\omega ^2\over {4\alpha }} \bigg )\bigg ]\ge 0,\quad \omega \in {\mathbb {R}};\) the previous inequality is satisfied if: \(\big [\exp \big ({\omega ^2/{4\alpha }}\big )\big ]/{(\omega ^2+\alpha ^2)^{3/2}}\ge B/{(2A\alpha ^2)}.\)

    The minimum value of the function \(g(\omega ;\alpha )=\big [\exp \big ({\omega ^2/{4\alpha }}\big )\big ]/{(\omega ^2+\alpha ^2)^{3/2}}\) is obtained for \(\omega =0\), if \(\alpha >6\): in this case, (57) is a c.f. if \(\alpha \le (2A)/B\); on the other hand, the minimum value of the function g is obtained for \(\omega =\sqrt{6\alpha -\alpha ^2}\) if: \(0<\alpha <6.\) In this last case, (57) is a c.f. if: \(e^{(6-\alpha )/4} \sqrt{\alpha }>(3B \sqrt{6})/A.\)

  • \(n=3.\) In this case, utilizing Eq. (47), the following three dimensional spectral density functions \(f_1\) and \(f_2\) can be obtained:

    $$\begin{aligned} f_1(\omega ;\alpha )={\alpha \over {\pi ^2(\omega ^2+\alpha ^2)^{2}}},\quad f_2(\omega ;\alpha ) = {1\over {8({\pi \alpha })^{3/2}}} \exp \bigg [-\bigg ({\omega ^2\over {4\alpha }} \bigg )\bigg ], \end{aligned}$$

    where \(\alpha >0, \quad \varvec{\omega }=(\omega _1,\omega _2,\omega _3), \omega =\sqrt{\omega _1^2+\omega _2^2+\omega _3^2}.\) Then

    $$\begin{aligned} C({x};A, B, \alpha ) = A \exp \big (-\alpha |x|\big )- B\exp \big (-\alpha x^2\big ), A>0, B>0, \end{aligned}$$
    (58)

    according to Corollary 3, is a c.f. if its Fourier transform is a spectral density function, i.e., \({A\alpha \over {\pi ^2(\omega ^2+\alpha ^2)^{2}}} - {B\over {8({\pi \alpha })^{3/2}}} \exp \bigg [-\bigg ({\omega ^2\over {4\alpha }} \bigg )\bigg ]\ge 0;\) the previous inequality is satisfied if: \(\big [\exp \big ({\omega ^2/{4\alpha }}\big )\big ]/{(\omega ^2+\alpha ^2)^{2}}\ge B\sqrt{\pi }/{(8A\alpha ^2\sqrt{\alpha })}.\)

    The minimum value of the function \(\delta (\omega ;\alpha )=\big [\exp \big ({\omega ^2/{4\alpha }}\big )\big ]/{(\omega ^2+\alpha ^2)^{2}}\) is obtained for \(\omega =0\), if \(\alpha >8\): in this case, (58) is a c.f. if \(\alpha \sqrt{\alpha }\le (8A)/(B\sqrt{\pi })\); on the other hand, the minimum value of the function \(\delta\) is obtained for \(\omega =\sqrt{8\alpha -\alpha ^2}\) if: \(0<\alpha <8.\) In this last case, (58) is a c.f. if: \(e^{(8-\alpha )/4} \sqrt{\alpha }>(8B \sqrt{\pi })/A.\)

3.4 Further properties of real covariance functions

Some properties related to isotropic/anisotropic and separable/nonseparable c.f.s are given hereafter.

  • If \(C_1\) and \(C_2\) in (6) are isotropic c.f.s, then C is an isotropic c.f.;

  • if \(C_1\) or \(C_2\) in (6) is an anisotropic c.f., then C is an anisotropic c.f.;

  • the sum of two anisotropic c.f.s could be an isotropic c.f.: let \(C_3(\mathbf{x} )=C_1(\Vert \mathbf{x} \Vert )-C_2(\mathbf{x} )\), where \(C_3\) satisfies Corollary 3, \(C_1\) is an isotropic c.f. and \(C_2\) is an anisotropic c.f.; then \(C(\mathbf{x} )=C_3(\mathbf{x} )+C_2(\mathbf{x} )\) is an isotropic c.f..

  • although the c.f. C in Eq. (11) is separable in \({\mathbb {R}}^n\), the c.f. C in (16) is non separable; anyway both models are characterized by non geometric anisotropy.

Real c.f.s can also be obtained as the Fourier transform of the difference of two spectral density functions, which are not both even functions. The following result is a generalization of Example 1.

Corollary 7

Let: \(f_1(\omega ; \alpha )=\exp (-\alpha |\omega |) P_n(\omega ),\quad f_2(\omega ; \alpha )=\exp (-\alpha |\omega |) Q_m(\omega ),\) be spectral density functions, \(\omega \in {\mathbb {R}}\), and let

$$\begin{aligned} f(\omega ;\alpha )\\=A_1 f_1(\omega ; \alpha )-A_2 f_2(\omega ; \alpha )=\exp (-\alpha |\omega |) \big (A_1 P_n(\omega )-A_2 Q_m(\omega )\big ), \end{aligned}$$

where \(P_n\) and \(Q_m\) are polynomials of degree n and m, respectively. Let \(C_1\) and \(C_2\) be the c.f.s corresponding, through Bochner’s theorem, to \(f_1\) and \(f_2\), respectively. If:

$$\begin{aligned} A_1 P_n(\omega )-A_2 Q_m(\omega ) &\ge 0, \quad \omega \in {\mathbb {R}}, \quad and\\ A_1 P_n(\omega )-A_2 Q_m(\omega )& = A_1 P_n(-\omega )-A_2 Q_m(-\omega ),\quad \omega \in {\mathbb {R}}, \end{aligned}$$

then \((A_1 f_1-A_2 f_2)\) is an even spectral density function and

$$\begin{aligned} C(x; \alpha )\\= A_1 C_1 (x; \alpha )- A_2 C_2 (x; \alpha )=\int _{{\mathbb {R}}} \exp (i{ \omega } {x})( A_1 f_1({\omega ;\alpha })- A_2 f_2({\omega ;\alpha })) d{ \omega }, \end{aligned}$$

is a real c.f., although \(C_1\) and \(C_2\) could be complex c.f.s if \(P_n(\omega )\ne P_n(-\omega )\), for some \(\omega \in {\mathbb {R}}\), and \(Q_m(\omega )\ne Q_m(-\omega )\), for some \(\omega \in {\mathbb {R}}\).

Note that it is necessary that the integers n and m in the previous corollary must be even numbers, otherwise \(f_1\) and \(f_2\) cannot be spectral density functions.

4 Covariance functions with negative values

The set of c.f.s with negative values is, of course, a subset of the real c.f.s. Apart from some particular cases which have been published in the literature, some examples of c.f.s characterized by negative values, are given hereafter. In general, for a complex c.f., only its real part is a c.f., hence it is possible to analyze the sign of the real part: some complex c.f.s whose real part is a covariance with negative values have been proposed. Some families regarding the difference of two real c.f.s, already described in subsections, characterized by negative values, have also been selected.

  1. 1.

    Yaglom (1987) furnished oscillatory c.f.s, based on the Bessel function of the first kind; on the other hand, Gneiting (2002a) proposed some families of c.f.s characterized by negative values through the turning bands technique.

  2. 2.

    C.f.s which are constructed as a linear combinations of c.f.s, with some negative weights, as described in Ma (2005) and in Gregori et al. (2008).

    In particular, in Ma (2005) linear combinations of two isotropic c.f.s of the same type were considered with further generalizations in the space-time domain.

    In Gregori et al. (2008) the following spatio-temporal class of the generalized sum of product models was analyzed, i.e.,

    $$\begin{aligned} C_{st}(\mathbf{x} ,u)= \sum _{j=1}^m k_j C_{sj}(\mathbf{x}) C_{tj}(u), \quad (\mathbf{x} ,u) \in {\mathbb {R}}^{n}\times {\mathbb {R}}, \end{aligned}$$
    (59)

    where \(C_{sj}\) and \(C_{tj}, j=1,\ldots ,m\), are continuous and integrable spatial and temporal c.f.s, respectively. Through a suitable analysis on the inf and sup of the ratios of the spectral density functions, the authors pointed out that the class (59) is still a class of c.f.s for some negative weights \(k_j.\)

  3. 3.

    The real part of a complex c.f. is a c.f.; there exist several classes of complex c.f.s whose real part is characterized by negative values in a subset of their domain, as pointed out in the following examples.

Example 17

Let’s consider, in \({\mathbb {R}}\), with \(\lambda>0, \quad p>0\), the following spectral density function (gamma distribution):

$$\begin{aligned} f(\omega ; \lambda , p) =\left\{ \begin{array}{ll} {\lambda ^p \over \Gamma (p)} \omega ^{p-1} e^{-\lambda \omega } &{} \omega >0\\ 0 &{} \omega \le 0, \\ \end{array} \right. \end{aligned}$$
(60)

and the corresponding c.f.:

$$\begin{aligned} C(x)=\Bigg ({\lambda \over {\lambda -i x}}\Bigg )^p. \end{aligned}$$
(61)

Please note that (60) is not an even spectral density function, hence (61) is a complex c.f.; a special case of (61), for \(p=2\) and \(\lambda =1\), is the following c.f.:

$$\begin{aligned} C(x) = {{1-x^2}\over {(x^2+1)^2}} +i {2x \over {(x^2+1)^2}}. \end{aligned}$$

Example 18

Let’s consider, in \({\mathbb {R}}\), with \(n \in \mathbb {N}_+\), the following spectral density function (\(\chi ^2\) distribution):

$$\begin{aligned} f(\omega ; n) =\left\{ \begin{array}{ll} {1\over {2^{n/2}\Gamma (n/2)}} \omega ^{n/2-1} e^{-\omega /2} &{} \omega >0\\ 0 &{} \omega \le 0, \\ \end{array} \right. \end{aligned}$$
(62)

and the corresponding c.f.:

$$\begin{aligned} C(x)=\Bigg ({1\over {1-2i x}}\Bigg )^{n/2}, \end{aligned}$$
(63)

which is very similar to (61). Even in this case, (62) is not an even spectral density function, hence (63) is a complex c.f.. A special case of (63), for \(n=4\), is the following c.f.:

$$C(x) = {{1-4x^2}\over {(4x^2+1)^2}} +i {4x \over {(4x^2+1)^2}}.$$

On the other hand, the difference of two real c.f.s can be also characterized by negative values, for suitable values of the vector of parameters, in a subset of their domain, as shown in the following example.

Example 19

Let’s consider, in \({\mathbb {R}}\), the following spectral density functions:

\(f_1(\omega ) = e^{-|\omega |} (1+|\omega |), \quad f_2(\omega ) =e^{-|\omega |},\) which are even functions; then,

$$\begin{aligned} C(x; A_1, A_2)= \int _{{\mathbb {R}}} \exp ({i \omega {x}}) \big [A_1 f_1(\omega )-A_2 f_2(\omega )\big ] d \omega \end{aligned}$$
(64)

is a real c.f. if and only if: \(\quad A_1 f_1(\omega )-A_2 f_2(\omega )\ge 0, \omega \in {\mathbb {R}};\) hence, (64) is a real c.f. if \(A_2<A_1.\) Then, the c.f. \(C(x; A_1, A_2)= {4 A_1 \over {(x^2+1)^2}} - {2 A_2 \over {(x^2+1)}}\) is negative in a subset of its domain, i.e., \(x<-\sqrt{{2A_1\over A_2} -1}, \quad x>\sqrt{{2A_1\over A_2} -1}.\)

  1. 4.

    C.f.s characterized by negative values can be selected among most of the families described in the examples proposed in Sect. 3.2 and the same formalism will be retained: the generic coordinate of the one dimensional space \({\mathbb {R}}\) will be denoted with \(x_k.\)

    In particular, in Example 7, it is easy to verify that the c.f. C in (30) is negative for all the values \(x_k \in {\mathbb {R}}\) which satisfy: \((\alpha _k-\beta _k) x_k^2 > \ln \big ({A_k\over {B_k}}\big ), \quad k=1,\ldots ,n;\)

    In Example 8, if \(\alpha _k>\beta _k\), it is easy to verify that the c.f. (32) is negative for the values \(x_k \in {\mathbb {R}}\) which satisfy: \((\alpha _k-\beta _k)|x_k|>\ln \big ({A_k\over {B_k}}\big ), k=1,\ldots ,n;\) on the other hand, if \(\alpha _k<\beta _k\), the c.f. (32) should be negative if: \((\beta _k-\alpha _k)|x_k|<\ln \big ({B_k\over A_k}\big );\) however, because of (34), \({B_k\over A_k}<{ {\alpha _k\over {\beta _k}}}<1;\) then, if \(\alpha _k<\beta _k, k=1,\ldots ,n,\) the c.f. (32) cannot ever be negative. In Example 9, the parametric family (35), in \({{\mathbb {R}}}\), becomes: \(C({x_k};A_k,B_k,\alpha _k,\beta _k)= {A_k \over { (x_k^2+\alpha _k^2)}}- {B_k \over { (x_k^2+\beta _k^2)}};\)

    in particular, this last family is characterized just by positive values in the whole domain if \(B_k<A_k\) and \(\beta _k>\alpha _k\), and by negative values in a subset of its domain if: \(1< {B_k\over A_k}<{\beta _k\over \alpha _k}.\) In Example 10, the parametric family (37), in \({{\mathbb {R}}}\), becomes:

    $$\begin{aligned} C({x_k}; A_k, B_k, \alpha _k)=\exp (-\alpha |x_k|)\bigg [ A_k - {{ \pi ^3 B_k}\over {8\alpha _k^3}} (1+\alpha _k|x_k|)\bigg ], \end{aligned}$$

    which is always characterized by negative values in a subset of its domain.

    In Example 11, the parametric family (38), in \({{\mathbb {R}}}\), becomes:

    $$\begin{aligned} C({x_k};A_k, A_k, \alpha _k,\beta _k)=A_k \exp (-\alpha _k |x_k|)-B_k \exp (-\beta _k x_k^2). \end{aligned}$$

    This last family is negative for the values \(x_k \in {\mathbb {R}}\) which satisfy: \((\beta _k x_k^2-\alpha _k |x_k|)<\ln \big ({B_k\over A_k}\big );\) taking into account the minimum value of the function: \(G({x_k};\alpha _k,\beta _k)= (\beta _k x_k^2-\alpha _k |x_k|),\) the following condition must be satisfied for the c.f., defined in (38), to be negative, i.e., \(\big ({\alpha _k^2\over {4\beta _k}}\big )>\ln \big ({A_k\over B_k}\big );\) hence, if the above condition is not satisfied, the c.f., defined in (38), cannot ever be negative. Moreover, if \(\alpha _k^2<4 \beta _k\) and \(0< {B_k\over A_k}<{\sqrt{\pi }\over \pi ^2}e^3, \quad \ln \big ({A_k\over B_k}\big )<{\alpha _k^2\over {4 \beta _k}}<1,\)

    the same family of c.f.s could assume negative values in a subset of its domain.

    In Example 12, if \(\alpha _k \beta _k>1\), the c.f.

    $$\begin{aligned} C({x_k}; A_k,B_k,\alpha _k, \beta _k)\\=A_k \exp \big (- \alpha _k |x_k| \big )- {B_k\over {(x_k^2+\beta _k^2)}}, \quad A_k>0, B_k>0, \end{aligned}$$

    is negative for the values \(x_k \in {\mathbb {R}}\) which satisfy: \((x_k^2+\beta _k^2)\exp (-\alpha _k|x_k|)<{B_k \over A_k};\)

    this last inequality can be easily satisfied for sufficiently high values of \(x_k,\) since

    $$\begin{aligned} \lim _{x_k\rightarrow \infty } \bigg [(x_k^2+\beta _k^2)\exp (-\alpha _k|x_k|)\bigg ]=0. \end{aligned}$$

    In Example 14 the difference between two isotropic Gaussian c.f.s has been considered in the Euclidean spaces \({\mathbb {R}}^2\) and \({\mathbb {R}}^3\), respectively: it is easy to verify that the c.f.s C in (49) and (52) are negative for all the values \(x \in {\mathbb {R}}\) which satisfy: \((\alpha -\beta ) x^2 > \ln \big ({A\over {B}}\big ).\)

    In Example 15 the difference between two isotropic exponential c.f.s has been considered in the Euclidean spaces \({\mathbb {R}}^2\) and \({\mathbb {R}}^3\), respectively. In particular, if \(\alpha <\beta\), the c.f.s C in (53) and (55) should be negative for the values \(x \in {\mathbb {R}}\) which satisfy: \((\beta -\alpha ) |x| <\ln \big ({B\over {A}}\big );\) however, this last inequality cannot ever be satisfied because \(1<{\beta \over \alpha }<{A\over B}\); on the other hand, if \(\alpha >\beta\), the c.f.s C in (53) and (55) are negative for the values \(x \in {\mathbb {R}}\) which satisfy: \((\alpha -\beta ) |x| > \ln \big ({A\over {B}}\big ).\)

5 Relevance and impact of the results

The motivation for this paper arises from the need to address various problems, whose importance concerns theoretical and practical aspects as specified hereafter:

  • a covariance is a complex valued function and its properties are well known and described in the literature. Moreover, it is also well known that, in general, the difference of two c.f.s is not a c.f. and this issue, in the literature, does not yet seem to have been addressed in detail. For this reason, the general problem concerning the difference of two c.f.s in the complex domain has been analyzed in this paper; hence, exploring the conditions such that the difference of two c.f.s is again a c.f. can be considered a further property. The results of this paper thus yield a general contribution to the theory of correlation, hence they cannot be restricted to the particular field of application. The above issue has been explored in the complex domain as well as in the subset of the real domain; in this latter case, the difference of isotropic c.f.s has also been discussed for the two and the three dimensional Euclidean space. In particular, a detailed analysis involving the parameters of these models has been proposed. Although the results provided in this paper represent a natural consequence of Bochner’s theorem, the significant aspects of the same results derive from properly specifying the characteristics of the spectral distribution function or of the spectral density function in Bochner’s representation.

  • All the families of c.f.s, which are constructed utilizing the standard models (Gaussian, general exponential and rational, which are non negative and non decreasing c.f.s) and by applying the usual and well known properties (such as linear combinations with non negative coefficients or products), preserve the main typical characteristics of the standard models, regardless of the values of their parameters, as formally shown hereafter.

    Let \(C_j\) be c.f.s such that \(C_j(x)\ge 0, x \in {\mathbb {R}}\); \(C_j'(x)<0, x>0, j=1,\ldots ,n\) and

    $$\begin{aligned} C_S(x)=\sum _{j=1}^n \lambda _j C_j (x), \quad C_P(x)=\prod _{j=1}^n C_j (x), \quad \lambda _j\ge 0, j=1,\ldots ,n; \end{aligned}$$
    (65)

    then:

    $$\begin{aligned} C_S(x)\ge & {} 0, x \in {\mathbb {R}}, \quad and \quad C_S'(x)<0, x>0,\\ C_P(x)\ge & {} 0, x \in {\mathbb {R}}\quad and \quad C_P'(x)<0, x>0; \end{aligned}$$

    in fact, a linear combination of these standard models (\(C_j\) in the above expressions) with non negative coefficients or their product, generate c.f.s which present the same behaviour of the same standard models: they are characterized by a systematic decreasing behaviour and are always non negative in their domain.

    On the other hand, there exist models constructed through the difference of standard models which behave differently by properly changing the values of the non negative parameters from which they depend. Hence, some special families of c.f.s obtained through the difference of two standard c.f.s provide a significant contribution to modeling wide classes of correlation structures. For example, the class of models defined in (32), i.e.,

    $$\begin{aligned} C({x_k};A_k, B_k, \alpha _k,\beta _k)\\=A_k \exp \big (-\alpha _k |x_k| \big ) - B_k \exp \big (-\beta _k|x_k|\big ), \quad A_k>0, B_k>0, \end{aligned}$$

    is flexible enough to describe several situations with reespect to the standard families of c.f.s as the ones defined in Eq. (65); indeed, the class (32) is always characterized by a linear behavior near the origin, it is always positive if: \({B_k\over A_k}<{ {\alpha _k\over {\beta _k}}}<1\); moreover, in this case, this class presents a systematically decreasing behavior for \(x_k>0\). On the other hand, if: \({B_k\over A_k}<{ {\beta _k\over {\alpha _k}}}<1\), the same class is negative for all the values \(x_k \in {\mathbb {R}}\) which satisfy: \((\alpha _k-\beta _k) |x_k|> \ln \big ({A_k\over {B_k}}\big )\). Hence, the same class behaves differently by properly changing the values of the parameters on which it depends.

    Likewise, the class of models defined in (35), i.e.,

    $$\begin{aligned} C(x_k;A_k, B_k, \alpha _k,\beta _k)= {A_k \over {(x_k^2+\alpha _k^2)}}- {B_k \over {(x_k^2+\beta _k^2)}}, \quad A_k>0, B_k>0, \end{aligned}$$

    is always characterized by a parabolic behavior near the origin and it is always positive if: \(\beta _k>\alpha _k\) and \(B_k<A_k\); moreover, in this case, this class presents a systematically decreasing behavior for \(x_k>0\). On the other hand, if: \(B_k>A_k\), the same class is negative for \(x_k \in {\mathbb {R}}\) which satisfy: \((B_k-A_k) x_k^2 > A_k \beta _k^2-B_k\alpha _k^2.\)

    Finally, some traditional families of c.f.s, characterized by damped oscillations and by negative values in subsets of their domain, have been constructed by multiplying standard models of c.f.s with a cosinusoidal c.f.; however, these last families of c.f.s are unable to behave differently by changing the values of the parameters on which they depend, as explained hereafter.

    Let \(C_D(x)=C(x) \cos (x)\), where C is a c.f. such that \(C(x)\ge 0, x \in {\mathbb {R}}\) and \(C'(x)<0, x>0.\) Then \(C'_D(x)=0\) if \(\tan (x)= {C'(x)\over C(x)};\) hence, there always exists \(x\in {\mathbb {R}}_+\) such that \(C_D\) changes its sign.

  • From a practical point of view, the various examples, concerning the difference of two c.f.s proposed in this paper, are flexible enough to be implemented and utilized in several applications pertaining to the complex domain, as well as to the subset of the real domain. In particular, the models obtained through this difference are characterized by different features: for example, they could be non negative in the whole domain, or could be characterized by negative values in a subset of their domain. Indeed, in several applications in biology, hydrology and spatio-temporal turbolence, c.f.s with negative values are often needed, as pointed out by various authors (Shkarofsky 1968; Levinson et al. 1984; Pomeroy et al. 2003; Xu et al. 2003a, b; Yakhot et al. 1989). The theory of complex random fields can give a significant contribution to several phenomena which often occur in ecological and environmental science: in particular, in meteorology, oceanography, signal analysis and geophysics. For this reason, in the very last years some efforts have been made in geostatistics utilizing complex formalism. In particular, an application to predict a wind field has been proposed by De Iaco and Posa (2016); moreover, a first attempt to utilize complex formalism in the spatio-temporal context has also been provided by Cappello et al. (2020): the complex c.f., indexed in time, is estimated and modeled for data, derived from high frequency radar systems, collected from 207 stations along the US East and Gulf Coast. In addition, in other particular scientific applications, flexible c.f.s models like the ones defined in Eqs. (32) and (35) are often required: in particular, in genetics and molecular biology the Price equation (Martini et al. 2016) utilizes c.f.s to provide a mathematical description of evolution and natural selection; in micrometeorology, the Eddy c.f. (Moncrieff et al. 2004) is a key atmospherics measurement technique; finally, c.f.s play a key role in feature extraction (Sahidullah and Kinnunen 2016), to capture the spectral variability of a signal.

6 Conclusions

In this paper, conditions for which the difference of two c.f.s in the complex domain, as well as in the subset of the real domain, is again a c.f. have been analyzed. This can be considered a further property which addresses some relevant issues: (a) it provides wider families of c.f.s which can be utilized to model further correlation structures; (b) it supplies families of c.f.s also characterized by negative values, utilized in several applications; (c) some special families of c.f.s obtained through the difference of two c.f.s provide a significant contribution to modeling wide classes of correlation structures because they are flexible enough to describe various behaviours as compared to the standard families. The c.f.s proposed in this paper can be utilized to model complex random fields, but also the subset of real random fields; in this latter case, some well known models have been considered. Moreover, in order to provide a complete analysis of this subject, some examples of the difference of two isotropic c.f.s have been also explored; in particular, for some of these examples the c.f.s involved in the difference could not belong to the same family. The results can be utilized in a flexible way in any dimensional domain.