Models for the difference of continuous covariance functions

A linear combination, with negative weights, of two continuous covariance functions has been analyzed by a few authors just for special cases and only in the real domain. However, a covariance is a complex valued function: for this reason, the general problem concerning the difference of two covariance functions in the complex domain needs to be analyzed, while it does not yet seem to have been addressed in the literature; hence, exploring the conditions such that the difference of two covariance functions is again a covariance function can be considered a further property. Therefore, this paper yields a contribution to the theory of correlation, hence the results cannot be restricted to the particular field of application. Starting from the difference of two complex covariance functions defined in one dimensional Euclidean space, wide families of models for the difference of two complex covariance functions can be built in any dimensional space, utilizing some well known properties. In particular, the difference of two real covariance functions has been considered; moreover, the difference between some special isotropic covariance functions has also been analyzed. A detailed analysis of the parameters of the models involved has been proposed; this kind of analysis opens a gate for modeling, in any dimensional space, the correlation structure of a peculiar class of complex valued random fields, as well as the subset of real valued random fields. Some relevant hints about how to construct the subset of real covariance functions characterized by negative values have also been given.


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, 2002Gneiting 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 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.

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 R n the Euclidean ndimensional space, with C the set of the complex numbers and i the imaginary unit.
Theorem 1 Bochner's theorem. Let C : R n ! C, be a Hermitian function, then C is a continuous c.f. if and only if it is of the form where F is a non decreasing and non negative bounded measure on R n .
The c.f. C in (1) can also be written as: 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. Cð0Þ ! 0; 2. CðÀxÞ ¼ CðxÞ; 3. jCðxÞj Cð0Þ; where the bar denotes complex conjugate. If the non negative spectral distribution function F is absolutely continuous, then there exists: f : R n À! R; such that: 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., If F is an absolutely continuous spectral distribution function, then F ðxÞ ¼ Cð0Þ À FðÀxÞ 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 distributioñ F is the conjugate c.f. of C, i.e., Hence, if C is a c.f., then 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 functionf ðxÞ ¼ f ðÀxÞ, corresponding to the conjugate spectral distributionF, is the conjugate c.f. of C of the spectral density functionf , i.e., An absolutely continuous spectral distribution function is said to be symmetric if it is equal to its conjugate, i.e., FðxÞ ¼ FðxÞ. 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., CðxÞ ¼ CðxÞ: 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.
where each component in the product is a c.f. in R; in particular, if F i is the spectral distribution function of C i in the representation (1), i ¼ 1; . . .; n, then C also has a representation as in (1), where • 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.

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 : R n ! C; i ¼ 1; 2 be c.f.s and define Cðx; KÞ ¼ AC 1 ðx; aÞ À BC 2 ðx; bÞ; x ¼ ðx 1 ; . . .; x n Þ 2 R n ; where A [ 0; B [ 0; a and b are vectors of parameters and K ¼ ðA; B; a; bÞ: Because of Bochner's theorem, two different cases need to be analyzed: • the spectral distribution functions F i ; i ¼ 1; 2, which define C i ; i ¼ 1; 2, respectively, are not absolutely continuous. In this case the corresponding c.f.s can be written as follows: is a c.f. if and only if, for some suitable values of the vector of parameters K and x 2 R n , ðAF 1 À BF 2 Þ is a non negative bounded function on R n ; • the spectral distribution functions F i ; i ¼ 1; 2, which define C i ; i ¼ 1; 2, respectively, are absolutely continuous. In this case, the corresponding c.f.s can be written as follows: is a c.f. if and only if, for some suitable values of the vector of parameters K and x 2 R n ; Af 1 ðx; aÞ À Bf 2 ðx; bÞ ! 0; Z R n ðAf 1 ðx; aÞ À Bf 2 ðx; bÞÞdx\ þ 1: The second property of (9) is always satisfied because ðAf 1 À Bf 2 Þ 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 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 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. it is a special case of Corollary 3 in R, 2. it is an application of Eq. (4).
In the Corollary 5, starting from separable models in C, the components of the real and imaginary part of the resulting c.f. are non separable and anisotropic.
As a consequence, C defined in Eq. (11) is a c.f.. • The spectral distribution functions F 1 ðx k ; a k Þ; F 2 ðx k ; b k Þ, which define C 1 ðx k ; a k Þ and C 2 ðx k ; b k Þ; k ¼ 1; . . .; n; respectively, are absolutely continuous. In this case, these c.f.s can be written as follows: with f j ðx k ; ÁÞ [ 0; x k 2 R and R R f j ðx k ; ÁÞdx k \1; j ¼ 1; 2; k ¼ 1; . . .; n; then, each is a c.f. if and only if, for some suitable values of the vector of parameters H and x k 2 R; k ¼ 1; . . .; n; 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 : R ! C; j ¼ 1; 2 be c.f.s and define where A ¼ ðA 1 ; . . .; A n Þ; B ¼ ðB 1 ; . . .; B n Þ; a ¼ ða 1 ; . . .; a n Þ and b ¼ ðb 1 ; . . .; b n Þ are vectors of parameters, x ¼ ðx 1 ; . . .; x n Þ 2 R n and H ¼ ðA; B; a; bÞ: Let the spectral distribution functions which define C 1 ðx k ; a k Þ and C 2 ðx k ; b k Þ; k ¼ 1; . . .; n; respectively, be absolutely continuous, with j ¼ 1; 2; k ¼ 1; . . .; n: In this case, the expression (16) can be written as follows: Then, (17) is a c.f. if and only if, for some suitable values of the vector of parameters H and x 2 R n ; 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 12ðkÞ ; k ¼ 1; . . .; n; then C is a separable c.f., where separability is referred to the complex domain C. However, the real and the imaginary components outcoming from the above product are non separable in 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 C. However, the real and the imaginary components of the resulting complex c.f. are non separable in 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 ðAC 1 À BC 2 Þ 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 ðAf 1 À Bf 2 Þ cannot be an even function, as a consequence, if conditions (9) are satisfied, ðAC 1 À BC 2 Þ 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 ðAf 1 À Bf 2 Þ could be an even function, as a consequence, if conditions (9) are satisfied, ðAC 1 À BC 2 Þ is a real c.f., as shown in the following example.
Example 1 Consider, in R, the following spectral density functions: which are not even functions, hence the corresponding c.f.s are complex functions. It is easy to verify that: f 1 ðxÞ À f 2 ðxÞ ¼ e Àjxj ; 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 ðxÞ C 2 ðxÞ; x 2 R n ; then ðC 2 À C 1 Þ is not, in general, a c.f., as shown in the following example.
• If C 1 and C 2 are real correlation functions, i.e., C 1 ð0Þ ¼ C 2 ð0Þ ¼ 1; then CðxÞ ¼ C 1 ðxÞ À C 2 ðxÞ cannot ever be a c.f. (except for the trivial case C 1 ðxÞ ¼ 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. 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.

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 R; i.e.,: where a ¼ ða 1 ; . . .; a n Þ; is a c.f., because ðF 1 À F 2 Þ is a spectral distribution function, i.e., it is non negative and a non decreasing function. Moreover, if ðF 1 À F 2 Þ is a non symmetric spectral distribution function, then ðC 1 À C 2 Þ is a complex c.f..
Corollary 6 is valid in R, however, utilizing Eqs. (4) and (5), it can be extended to any dimensional space 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 R will be denoted with x k ; k ¼ 1; . . .; n. In this way, starting from the one dimensional space, covariance models in 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 R, the following spectral density functions, with 0\b k \a k ; 0\B k \A k ; x k 2 R; k ¼ 1; . . .; n: which are not even functions; since: According to Corollaries 4 and 5, the following functions: Example 4 Let's consider, in R, the following spectral density functions, with 0\a k \b k ; 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: then the following difference: is a complex c.f..
According to Corollaries 4 and 5, the following functions: ...;B n Þ;a ¼ ða 1 ;...;a n Þ; b ¼ ðb 1 ;...;b n Þ and H ¼ ðA;B;a;bÞ are vectors of parameters, x ¼ ðx 1 ;...;x n Þ 2 R n ; and Example 5 Let's consider, in R, the spectral density functions defined in the Eqs. (23) and (22) . . .; n; note that f 1 and f 2 are not even functions and define: this last function is a spectral density function if 0\B k \A k and If C 1 and C 2 are the corresponding c.f.s of f 1 and f 2 , respectively, then the following difference: with According to Corollaries 4 and 5, the following functions: . . .; a n Þ; b ¼ ðb 1 ; . . .; b n Þ and H ¼ ðA; B; a; bÞ are vectors of parameters, x ¼ ðx 1 ; . . .; x n Þ 2 R n ; and Example 6 Let's consider, in R, the spectral density functions 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 this last function is a spectral density function if are the corresponding c.f.s of f 1 and f 2 , respectively, then the following difference: is a complex c.f.. According to Corollaries 4 and 5, the following functions: . . .; A n Þ; B ¼ ðB 1 ; . . .; B n Þ; a ¼ ða 1 ; . . .; a n Þ; b ¼ ðb 1 ; . . .; b n Þ and H ¼ ðA; B; a; bÞ are vectors of parameters,

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 ¼ 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 be Gaussian c.f.s; these models are integrable, then their Fourier transforms f 1 and f 2 are, respectively: x k 2 R; k ¼ 1; . . .; n: Then according to Corollaries 3 and 4, is a c.f., for each k ¼ 1; . . .; n, if its Fourier transform is a spectral density function, i.e., A k f 1 ðx k ; a k Þ À B k f 2 ðx k ; b k Þ ! 0; x k 2 R; the previous inequality is satisfied if: Example 8 Given the following exponential c.f.s: then, C 1 and C 2 are integrable and is a c.f. if its Fourier transform f is a spectral density function, i.e., in particular, (33) is satisfied if: According to the values of the parameters a k and b k ; k ¼ 1; . . .; n, some special situations can be analyzed: 1. a k [ b k ; k ¼ 1; . . .; n; in this case, it is easy to verify that the minimum value of the function: ; as a consequence, (33) is a spectral density function if: (33) is a spectral density function if: Example 9 Given the following c.f.s, then C 1 and C 2 are integrable and is a c.f. if its Fourier transform f is a spectral density function, i.e., in particular, (36) Example 10 Given the following c.f.s: . . .; n; then, C 1 and C 2 are integrable and is a c.f. if its Fourier transform f is a spectral density function, i.e., Note that the maximum value of the function Note that: cannot be a c.f. because its Fourier transform: cannot be a spectral density function, since the function: does not present a minimum value. Example 11 Given the following c.f.s k ¼ 1; . . .; n; then C 1 and C 2 are integrable and is a c.f., for each k ¼ 1; . . .; n, if the corresponding Fourier transform f is a spectral density function, i.e., According to the values of the parameters a k and b k ; k ¼ 1; . . .; n, some special situations can be analyzed: 1. a 2 k À 4b k [ 0; k ¼ 1; . . .; n; in this case, it is easy to verify that the minimum value of the function: is obtained for x Ã k ¼ 0; then, taking into account the inequalities a 2 k À 4b k [ 0; k ¼ 1; . . .; n, f is a spectral density function, as a consequence C defined in (38) 2. a 2 k À 4b k \0; k ¼ 1; . . .; n; in this case, the minimum value of the function: jðx k ; a k ; b k Þ ¼ ; then, taking into account the inequalities a 2 k À 4b k \0; k ¼ 1; . . .; n, f is a spectral density function, as a consequence C defined in (38) cannot be a c.f. because its Fourier transform: cannot be a spectral density, since the function: does not present an absolute maximum value.
Example 12 Given the c.f.s k ¼ 1; . . .; n; then C 1 and C 2 are integrable and is a c.f. if the corresponding Fourier transform f is a spectral density function, i.e., According to the values of the parameters a k and b k ; k ¼ 1; . . .; n, some special situations can be analyzed: 1. a k b k [ 1; k ¼ 1; . . .; n; in this case, the minimum value of the function is obtained for x Ã k ¼ 0; hence, f is a spectral density function, as a consequence C defined in (39) is a c.f. if: . . .; n; in this case, there exist two minimum values for the function (40): : By comparing the values of the function v, defined in (40), computed in x Ã k and x ÃÃ k , i.e., it results that (41) is satisfied if: Inequality (42) is satisfied for 0:8\a k b k \1, where as for 0\a k b k \0:8 it results vðx ÃÃ k ; a k ; b k Þ À vðx Ã k ; a k ; b k Þ\0: Hence, according to the previous results: cannot be a c.f. because its Fourier transform: cannot be a spectral density function, since the function does not present an absolute maximum value.
Example 13 Given the following c.f.s, k ¼ 1; . . .; n, then C 1 and C 2 are integrable and is a c.f. if its Fourier transform f is a spectral density function, i.e., which is equivalent to the following inequality: the minimum value of the function sðx k ; a k ; b k Þ ¼ exp (44) is a spectral density function if: . . .; n: Note that: cannot be a c.f. because its Fourier transform: cannot be a spectral density function, since the following inequality is not satisfied, because the function s 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: are real c.f.s, both defined in R n ; where A ¼ ðA 1 ; . . .; A n Þ; B ¼ ðB 1 ; . . .; B n Þ; a ¼ ða 1 ; . . .; a n Þ; b ¼ ðb 1 ; . . .; b n Þ and H ¼ ðA; B; a; bÞ are vectors of parameters, x ¼ ðx 1 ; . . .; x n Þ 2 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.  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.

Difference of isotropic covariance functions
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 ¼ 2 and n ¼ 3, which are the most relevant cases in the applications: To check whether or not a given function C is an n-dimensional isotropic c.f. (for n ¼ 2 and n ¼ 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 R n , for any n 2 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 R 2 and R 3 , respectively.
• n ¼ 2. In the Example 7, given the one dimensional c.f.s C 1 ðx k Þ and C 2 ðx k Þ and the corresponding one dimensional spectral density functions f 1 ðx k Þ and f 2 ðx k Þ, by utilizing Eq. (46), the two dimensional spectral density functions f 1 ðxÞ and f 2 ðxÞ can be obtained: Cðx; A; B; a; bÞ according to Corollary 3, is a c.f. if its Fourier transform is a spectral density function, i.e., Af 1 ðx; aÞ À Bf 2 ðx; bÞ the previous inequality is satisfied if: • n ¼ 3. In this case, utilizing Eq. (47), the following three dimensional spectral density functions f 1 ðxÞ and f 2 ðxÞ can be obtained: ð51Þ the previous inequality is satisfied if: Example 15 In the following example, the difference between two isotropic exponential c.f.s has been considered in the Euclidean spaces R 2 and R 3 ; respectively.
• n ¼ 2. In the Example 8, given the c.f.s C 1 ðx k Þ and C 2 ðx k Þ and the corresponding one dimensional spectral density functions f 1 ðx k Þ and f 2 ðx k Þ, by utilizing Eq. (46), the two dimensional spectral density functions f 1 ðxÞ and f 2 ðxÞ can be obtained: according to Corollary 3, is a c.f. if its Fourier transform is a spectral density function, i.e., Aa in particular, • n ¼ 3: In this case, utilizing Eq. (47), the following three dimensional spectral density functions f 1 and f 2 can be obtained: ð55Þ according to Corollary 3, is a c.f. if its Fourier transform is a spectral density function, i.e., in particular, If a\b, then (56) Example 16 In the following example, the difference between an isotropic exponential and an isotropic Gaussian c.f.s has been considered in R 2 and R 3 , respectively.
• n ¼ 2. In the Examples 8 and 7, given the one dimensional c.f.s C 1 ðx k Þ and C 2 ðx k Þ for the exponential and the Gaussian models, respectively, with the corresponding one dimensional spectral density functions f 1 ðx k Þ and f 2 ðx k Þ, by utilizing Eq. (46), the two dimensional spectral density functions f 1 ðxÞ and f 2 ðxÞ, with a ¼ b, can be obtained: according to Corollary 3, is a c.f. if its Fourier transform is a spectral density function, i.e., Aa The minimum value of the function gðx; aÞ ¼ Â exp À x 2 =4a ÁÃ =ðx 2 þ a 2 Þ 3=2 is obtained for x ¼ 0, if a [ 6: in this case, (57) is a c.f. if a ð2AÞ=B; on the other hand, the minimum value of the function g is obtained for x ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6a À a 2 p if: 0\a\6: In this last case, (57) is a c.f. if: e ð6ÀaÞ=4 ffiffi ffi a p [ ð3B ffiffi ffi 6 p Þ=A: • n ¼ 3: In this case, utilizing Eq. (47), the following three dimensional spectral density functions f 1 and f 2 can be obtained:

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 ðxÞ ¼ C 1 ðkxkÞ À C 2 ðxÞ, where C 3 satisfies Corollary 3, C 1 is an isotropic c.f. and C 2 is an anisotropic c.f.; then CðxÞ ¼ C 3 ðxÞ þ C 2 ðxÞ is an isotropic c.f.. • although the c.f. C in Eq. (11) is separable in 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 ðx; aÞ ¼ expðÀajxjÞP n ðxÞ; f 2 ðx; aÞ ¼ expðÀajxjÞQ m ðxÞ; be spectral density functions, x 2 R, and let f ðx; aÞ 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: A 1 P n ðxÞ À A 2 Q m ðxÞ ! 0; x 2 R; and A 1 P n ðxÞ À A 2 Q m ðxÞ ¼ A 1 P n ðÀxÞ À A 2 Q m ðÀxÞ; x 2 R; then ðA 1 f 1 À A 2 f 2 Þ is an even spectral density function and Cðx; aÞ ¼ A 1 C 1 ðx; aÞ À A 2 C 2 ðx; aÞ is a real c.f., although C 1 and C 2 could be complex c.f.s if P n ðxÞ 6 ¼ P n ðÀxÞ, for some x 2 R, and Q m ðxÞ 6 ¼ Q m ðÀxÞ, for some x 2 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.

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. 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. 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 spatiotemporal class of the generalized sum of product models was analyzed, i.e., where C sj and C tj ; j ¼ 1; . . .; 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. 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 R, with k [ 0; p [ 0, the following spectral density function (gamma distribution): and the corresponding c.f.: 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 ¼ 2 and k ¼ 1, is the following c.f.: Example 18 Let's consider, in R, with n 2 N þ , the following spectral density function (v 2 distribution): and the corresponding c.f.: 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 ¼ 4, is the following c.f.: 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.
is negative in a subset of its domain, i.e., x\ À ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 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 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 2 R which satisfy: . . .; n; In Example 8, if a k [ b k , it is easy to verify that the c.f. (32) is negative for the values x k 2 R which satisfy: . . .; n; on the other hand, if a k \b k , the c.f. (32) should be negative if: ; however, because of (34), ; then, if a k \b k ; k ¼ 1; . . .; n; the c.f. (32) cannot ever be negative. In Example 9, the parametric family (35), in R, becomes: in particular, this last family is characterized just by positive values in the whole domain if B k \A k and b k [ a k , and by negative values in a subset of its In Example 10, the parametric family (37), in R, becomes: which is always characterized by negative values in a subset of its domain. In Example 11, the parametric family (38), in R, becomes: This last family is negative for the values x k 2 R which satisfy: ðb k x 2 k À a k jx k jÞ\ ln ; taking into account the minimum value of the function: Gðx k ; a k ; b k Þ ¼ ðb k x 2 k À a k jx k jÞ; the following condition must be satisfied for the c.f., defined in (38) the same family of c.f.s could assume negative values in a subset of its domain.
In Example 12, if a k b k [ 1, the c.f.
is negative for the values x k 2 R which satisfy: ðx 2 k þ b 2 k Þ expðÀa k jx k jÞ\ B k A k ; this last inequality can be easily satisfied for sufficiently high values of x k ; since In Example 14 the difference between two isotropic Gaussian c.f.s has been considered in the Euclidean spaces R 2 and R 3 , respectively: it is easy to verify that the c.f.s C in (49) and (52)  : In Example 15 the difference between two isotropic exponential c.f.s has been considered in the Euclidean spaces R 2 and R 3 , respectively. In particular, if a\b, the c.f.s C in (53) and (55)

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 ðxÞ ! 0; x 2 R; C 0 j ðxÞ\0; x [ 0; j ¼ 1; . . .; n and then: C S ðxÞ ! 0; x 2 R; and C 0 S ðxÞ\0; x [ 0; C P ðxÞ ! 0; x 2 R and C 0 P ðxÞ\0; x [ 0; 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., 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: ; moreover, in this case, this class presents a systematically decreasing behavior for x k [ 0. On the other hand, if: B k A k \ b k a k \1, the same class is negative for all the values x k 2 R which satisfy: . 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., is always characterized by a parabolic behavior near the origin and it is always positive if: b k [ a 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 2 R which satisfy: ðB k À A k Þx 2 k [ A k b 2 k À B k a 2 k : 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.
• 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.

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.
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