The Gauss hypergeometric covariance kernel for modeling second-order stationary random fields in Euclidean spaces: its compact support, properties and spectral representation

Abstract

This paper presents a parametric family of compactly-supported positive semidefinite kernels aimed to model the covariance structure of second-order stationary isotropic random fields defined in the d-dimensional Euclidean space. Both the covariance and its spectral density have an analytic expression involving the hypergeometric functions ${}_2F_1$ and ${}_1F_2$, respectively, and four real-valued parameters related to the correlation range, smoothness and shape of the covariance. The presented hypergeometric kernel family contains, as special cases, the spherical, cubic, penta, Askey, generalized Wendland and truncated power covariances and, as asymptotic cases, the Matérn, Laguerre, Tricomi, incomplete gamma and Gaussian covariances, among others. The parameter space of the univariate hypergeometric kernel is identified and its functional properties—continuity, smoothness, transitive upscaling (montée) and downscaling (descente)—are examined. Several sets of sufficient conditions are also derived to obtain valid stationary bivariate and multivariate covariance kernels, characterized by four matrix-valued parameters. Such kernels turn out to be versatile, insofar as the direct and cross-covariances do not necessarily have the same shapes, correlation ranges or behaviors at short scale, thus associated with vector random fields whose components are cross-correlated but have different spatial structures.

Appendices

Appendix 1: Technical definitions and lemmas

Definition 1

(Montée and descente) For $k \in {\mathbb {N}}$, $k<d$, the transitive upgrading or montée of order k is the operator ${\mathfrak {M}}_k$ that transforms an isotropic covariance in ${\mathbb {R}}^d$ into an isotropic covariance in ${\mathbb {R}}^{d-k}$ with the same radial spectral density (Matheron 1965). The reciprocal operator is the transitive downgrading (descente) of order k and is denoted as ${\mathfrak {M}}_{-k}$.

Definition 2

(Conditionally negative semidefinite matrix) A symmetric real-valued matrix ${\varvec{A}}=[a_{ij}]_{i,j=1}^p$ is conditionally negative semidefinite if, for any vector $\varvec{\omega }=[\omega _{i}]_{i=1}^p$ in ${\mathbb {R}}^p$ whose components add to zero, one has $\sum _{i=1}^p \sum _{j=1}^p \omega _i \, a_{ij} \omega _j \le 0$.

Example 1

Examples of conditionally negative semidefinite matrices include the all-ones matrix ${\varvec{1}}$ or the matrix ${\varvec{A}}=[a_{ij}]_{i,j=1}^p$ with

$$\begin{aligned} a_{ij} = \frac{\eta _{i}+\eta _{j}}{2} + \psi ({\varvec{s}}_i,{\varvec{s}}_j), \end{aligned}$$

for any $\eta _{1}, \ldots , \eta _{p}$ in ${\mathbb {R}}$, ${\varvec{s}}_1, \ldots , {\varvec{s}}_p$ in ${\mathbb {R}}^d$, and variogram $\psi$ on ${\mathbb {R}}^d \times {\mathbb {R}}^d$ (Matheron 1965; Chilès and Delfiner 2012). Also, the set of $p \times p$ conditionally negative semidefinite matrices is a closed convex cone, so that the product of a conditionally negative semidefinite matrix with a nonnegative constant, the sum of two conditionally negative semidefinite matrices, or the limit of a convergent sequence of conditionally negative semidefinite matrices are still conditionally negative semidefinite.

Lemma 1

(Berg et al. (1984)) A symmetric real-valued matrix ${\varvec{A}}=[a_{ij}]_{i,j=1}^p$ is conditionally negative semidefinite if and only if $[\exp (- t \, a_{ij})]_{i,j=1}^p$ is positive semidefinite for all $t \ge 0$.

Definition 3

(Multiply monotone function) For $q \in {\mathbb {N}}$, a q-times differentiable function $\varphi$ on ${\mathbb {R}}_{\ge 0}$ is $(q+2)$-times monotone if $(-1)^k \varphi ^{(k)}$ is nonnegative, nonincreasing and convex for $k=0, \ldots ,q$. A 1-time monotone function is a nonnegative and nonincreasing function on ${\mathbb {R}}_{\ge 0}$ (Williamson 1956).

Lemma 2

(Williamson (1956)) A $(q+2)$-times monotone function, $q \ge -1$, admits the expression

$$\begin{aligned} \varphi (x) = \int _0^{+\infty } (1 - t \, x)_+^{q+1} \,\nu (\text {d}t), \qquad x \in {\mathbb {R}}_{\ge 0}, \end{aligned}$$

(15)

where $\nu$ is a nonnegative measure.

Example 2

Examples of $(q+2)$-times monotone functions include the truncated power function $x \mapsto b+(1 - \frac{x}{a})_+^{\eta }$ with $a>0$, $b \ge 0$ and $\eta \ge q+1$, the completely monotone functions, and positive mixtures and products of such functions.

Lemma 3

Let $q \in {\mathbb {N}}$, $\alpha , \beta$, $\gamma \in {\mathbb {R}}_{>0}$ and $\psi _1$ a positive function in ${\mathbb {R}}_{\ge 0}$ whose derivative is $(q+1)$-times monotone. Then, the function $\Phi _1: {\mathbb {R}}^{2q+1} \rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} {\it{\Phi}} _1({{\varvec{x}}}) = {}_1 F_2\left( \alpha ;\beta ,\gamma ; -\psi _1(\Vert {{\varvec{x}}} \Vert ) \right) , \quad {{\varvec{x}}} \in {\mathbb {R}}^{2q+1}, \end{aligned}$$

(16)

is a stationary isotropic covariance kernel in ${\mathbb {R}}^{2q+1}$ if $(\alpha +q+2,\beta +q+2,\gamma +q+2) \in \mathcal {P}_0$.

Example 3

Examples of functions $\psi _1$ satisfying the conditions of Lemma 3 include the integrated truncated power function $\psi _1(x)= b \, x + c-(1 - \frac{x}{a})_+^{\eta +1}$ ($a>0$, $b>0$, $c > 1$ and $\eta \ge q$) and the Bernstein functions (positive primitives of completely monotone functions), e.g. (Schilling et al. 2010):

• (1) $\psi _1(x) = 1+\log \left( 1+\frac{x}{b}\right)$ with $b > 0$;

• (2) $\psi _1(x) = \left( 1+b \, x^\eta \right) ^\theta$ with $b > 0$, $\eta \in ]0,1]$ and $\theta \in ]0,1]$;

• (3) $\psi _1(x) = 1+x \, (x+b)^{-\eta }$ with $b > 0$ and $\eta \in ]0,1]$.

Lemma 4

Let $q^{\prime } \in {\mathbb {N}}$, $\gamma > 0$, $x > 0$ and $\psi _2$ a positive function in ${\mathbb {R}}_{\ge 0}$ upper bounded by $\alpha _{\max }=\frac{2\gamma -1}{4}$ and whose derivative is $(q^{\prime }+1)$-times monotone. Then, the function $\Phi _2: {\mathbb {R}}^{2q^{\prime }+1} \rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} \Phi _2({\varvec{y}}) = {}_1 F_2\left( \psi _2(\Vert {\varvec{y}}\Vert );\psi _2(\Vert {\varvec{y}}\Vert )+1,\gamma ;-x \right) , \quad {\varvec{y}} \in {\mathbb {R}}^{2q^{\prime }+1}, \end{aligned}$$

(17)

is a stationary isotropic covariance kernel in ${\mathbb {R}}^{2q^{\prime }+1}$.

Lemma 5

Let $q, q^{\prime } \in {\mathbb {N}}$, $\gamma > 0$, $\psi _1$ a positive function in ${\mathbb {R}}_{\ge 0}$ with a $(q+1)$-times monotone derivative, and $\psi _2$ a positive function in ${\mathbb {R}}_{\ge 0}$ upper bounded by $\alpha _{\max }=\frac{2\gamma -1}{4}$ and with a $(q^{\prime }+1)$-times monotone derivative. Then, the function $\Phi : {\mathbb {R}}^{2q+1} \times {\mathbb {R}}^{2q^{\prime }+1} \rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned}&\Phi ({\varvec{x}},{\varvec{y}}) = \frac{1}{\psi _1(\Vert {\varvec{x}}\Vert )} {}_1 F_2\left( \psi _2(\Vert {\varvec{y}}\Vert );\psi _2(\Vert {\varvec{y}}\Vert )+1,\gamma ;-\psi _1(\Vert {\varvec{x}}\Vert ) \right) ,\nonumber \\&\quad {\varvec{x}} \in {\mathbb {R}}^{2q+1}, {\varvec{y}} \in {\mathbb {R}}^{2q^{\prime }+1}, \end{aligned}$$

(18)

is positive semidefinite in ${\mathbb {R}}^{2q+1} \times {\mathbb {R}}^{2q^{\prime }+1}$ if $(\alpha +q+3,\alpha +q+4,\gamma +q+3) \in \mathcal {P}_0$.

Appendix 2: Proofs

Proof of Theorem 2

Let $(\alpha ,\beta ,\gamma ) \in \mathcal {P}_d$. As the complex extension of the generalized hypergeometric function $x \mapsto {}_1F_2(\alpha ,\beta ,\gamma ,x)$, $x \in {\mathbb {C}}$, is an entire function not identically equal to zero, its zeroes (if they exist) are isolated. It follows that there exists a nonempty open interval $I \subseteq {\mathbb {R}}$ such that ${}_1F_2(\alpha ,\beta ,\gamma ,x)$ does not vanish, hence is positive, for all $x \in I$. Accordingly, the support of the spectral density (8) contains a nonempty open set of ${\mathbb {R}}^d$, which implies that the associated covariance kernel is positive definite (Dolloff et al. 2006). $\square$

Proof of Theorem 3

The claim stems from the fact that $g_{d-k}(\cdot ; a,\alpha -\frac{k}{2},\beta -\frac{k}{2},\gamma -\frac{k}{2})$ is the same as $g_d(\cdot ; a,\alpha ,\beta ,\gamma )$ and that $(\alpha -\frac{k}{2},\beta -\frac{k}{2},\gamma -\frac{k}{2}) \in \mathcal {P}_{d-k}$ as soon as $(\alpha ,\beta ,\gamma ) \in \mathcal {P}_{d}$. $\square$

Proof of Theorem 4

The proof is analog to that of Theorem 3, with the additional restriction to ensure that the extended covariance remains valid in ${\mathbb {R}}^{d+k}$. $\square$

Proof of Theorem 5

The continuity and differentiability with respect to r stem from the fact that the Gauss hypergeometric function $x \mapsto {}_2F_1(a_1,a_2;b_1;x)$ with $b_1-a_1-a_2>0$ is continuous on the interval [0, 1], equal to 1 at $x=0$, and infinitely differentiable on ]0, 1[. One deduces the continuity and differentiability with respect to a by noting that, for fixed $\alpha$, $\beta$ and $\gamma$, $g_d(r; a,\alpha ,\beta ,\gamma )$ only depends on $\frac{r}{a}$. Finally, the continuity and differentiability with respect to $\alpha$, $\beta$ and $\gamma$ stem from the fact that the exponential function of base $\left( 1-(\frac{r}{a})^2\right) _+$ and the gamma function are infinitely differentiable wherever they are defined, and the hypergeometric function ${}_2F_1$ is an entire function of its parameters. $\square$

Proof of Theorem 6

From (10), it is seen that $r \mapsto g_d(r; a,\alpha ,\beta ,\gamma )$ is of the order of $(1-\frac{r}{a})^{\beta +\gamma - \alpha -\frac{d}{2}-1}$ as $r \rightarrow a^-$, while it is identically zero for $r \rightarrow a^+$. Hence, this function is k times differentiable (with zero derivatives of order $1, 2, \ldots , k$) at $r=a$, if, and only if, $\beta - \alpha +\gamma > k+\frac{d}{2}+1$. $\square$

Proof of Theorem 7

Using formula E.2.3 of Matheron (1965), one obtains, for $\alpha -\frac{d}{2} \not \in {\mathbb {N}}$:

$$\begin{aligned}&g_d(r; a,\alpha ,\beta ,\gamma ) = {}_2F_1\left( \frac{d}{2}-\gamma +1,\frac{d}{2}-\beta +1;\frac{d}{2}-\alpha +1;\frac{r^2}{a^2} \right) \nonumber \\&\quad +\frac{\varGamma (\frac{d}{2}-\alpha ) \varGamma (\beta -\frac{d}{2}) \varGamma (\gamma -\frac{d}{2}) }{ \varGamma (\alpha -\frac{d}{2}) \varGamma (\beta -\alpha )\varGamma (\gamma -\alpha )} \left( \frac{r}{a}\right) ^{2\alpha -d} \nonumber \\&\quad \times {}_2F_1\left( \alpha -\beta +1,\alpha -\gamma +1;\alpha -\frac{d}{2}+1;\frac{r^2}{a^2} \right) , \nonumber \\&\quad 0 \le r < a. \end{aligned}$$

(19)

The right-hand side of (19) is a power series of $r^2$, plus a power series of $r^2$ (with a constant nonzero term) multiplied by $r^{2\alpha -d}$. Since $2\alpha -d$ is not an even integer, $r \mapsto g_d(r; a,\alpha ,\beta ,\gamma )$ turns out to be k times differentiable at $r=0$ if, and only if, $\alpha >\frac{k+d}{2}$. If $\alpha -\frac{d}{2} \in {\mathbb {N}}$, then formula E.2.4 of Matheron (1965) shows that $r \mapsto g_d(r; a,\alpha ,\beta ,\gamma )$ is a power series of $r^2$ plus a power series of $r^2$ (with a constant nonzero term) multiplied by $r^{2\alpha -d} \log (\frac{r}{a})$, and the same conclusion prevails: $r \mapsto g_d(r; a,\alpha ,\beta ,\gamma )$ is k times differentiable at $r=0$ if, and only if, $\alpha >\frac{k+d}{2}$. $\square$

Proof of Theorem 8

Using an integral representation of the Gauss hypergeometric function ${}_2F_1$ (Gradshteyn and Ryzhik 2007, formula 9.111), the restriction of the radial function $g_d$ on the interval [0, a] can be written as follows:

$$\begin{aligned}&g_d(r; a,\alpha ,\beta ,\gamma ) \nonumber \\&\quad = \frac{\varGamma (\gamma -\frac{d}{2}) }{\varGamma (\gamma -\alpha ) \varGamma (\alpha -\frac{d}{2})} \left( 1 - \frac{r^2}{a^2} \right) ^{\beta - \alpha +\gamma -\frac{d}{2}-1} \nonumber \\&\quad \times \int _0^1 t^{\gamma -\alpha -1} (1-t)^{\alpha -\frac{d}{2}-1} \left( 1+\frac{t}{1-t} \frac{r^2}{a^2} \right) ^{\alpha -\beta } \text {d}t, \quad r \in [0,a]. \end{aligned}$$

(20)

Accordingly, on [0, a], $r \mapsto g_d(r; a,\alpha ,\beta ,\gamma )$ is a beta mixture of functions of the form $r \mapsto ( 1 - \frac{r^2}{a^2})^{\beta - \alpha +\gamma -\frac{d}{2}-1}$ multiplied by functions of the form $r \mapsto (1 + \frac{t}{1-t} \frac{r^2}{a^2})^{\alpha -\beta }$, with $t \in ]0,1[$, $a>0$ and $(\alpha ,\beta ,\gamma )\in \mathcal {P}_d$. Since all these functions are nonnegative and decreasing on [0, a], so is $r \mapsto g_d(r; a,\alpha ,\beta ,\gamma )$.

The monotonicity in r implies the monotonicity in a, insofar as $g_d(r; a,\alpha ,\beta ,\gamma )$ only depends on $\frac{r}{a}$ for fixed $\alpha$,$\beta$ and $\gamma$.

For $r=0$, the mapping $\gamma \mapsto g_d(0; a,\alpha ,\beta ,\gamma )$ is identically equal to 1, hence constant in $\gamma$. Let now $r>0$ and consider the integral representation (9) as a function of $r = \Vert {\varvec{h}}\Vert$, a, $\alpha$, $\beta$ and $\gamma$. Based on the dominated convergence theorem, this function can be differentiated under the integral sign with respect to parameter $\gamma$, which leads to:

$$\begin{aligned}&\frac{\partial g_d(r; a,\alpha ,\beta ,\gamma )}{\partial \gamma } \nonumber \\&\quad = \frac{ \varGamma (\beta - \frac{d}{2})}{ \varGamma (\alpha -\frac{d}{2}) \varGamma (\beta -\alpha )} \int _{0}^1 t^{\alpha -\gamma } (1-t)_+^{\beta -\alpha -1}\nonumber \\&\qquad \left( t - \left( \frac{r}{a}\right) ^2 \right) _+^{\gamma - \frac{d}{2} - 1} \ln \left[ \left( 1-\frac{r^2}{t a^2} \right) _+\right] \text {d}t. \end{aligned}$$

(21)

As $r>0$, the above integral is convergent because the integrand is nonzero if, and only if, t belongs to $]\frac{r^2}{a^2},1[$ (empty interval if $r\ge a$), so the zero lower bound of the integral can be replaced by $\min \{1,\frac{r^2}{a^2}\}$. The partial derivative (21) is therefore always negative (if $0<r<a$) or zero (if $r \ge a$), implying that $\gamma \mapsto g_d(r; a,\alpha ,\beta ,\gamma )$ is decreasing or constant in $\gamma$, respectively. The same result holds by substituting $\beta$ for $\gamma$ owing to the symmetry of the ${}_2F_1$ function. $\square$

Proof of Theorem 9

For $(\alpha ,\beta ,\gamma ) \in \mathcal {P}_d$, the radial part of ${\mathfrak {M}}_k (G_d(\cdot ,a,\alpha ,\beta ,\gamma ))$ is the Hankel transform of order $d-k$ of ${\widetilde{g}}_d(\cdot ,a,\alpha ,\beta ,\gamma )$. From (3), one has

$$\begin{aligned} {\mathfrak {M}}_k( G_d(\cdot ,a,\alpha ,\beta ,\gamma )) = \frac{\zeta _d(a,\alpha ,\beta ,\gamma )}{\zeta _{d-k}(a,\alpha ,\beta ,\gamma )} G_{d-k}(\cdot ,a,\alpha ,\beta ,\gamma ). \end{aligned}$$

Since $\mathcal {P}_d \subset \mathcal {P}_{d-k}$, ${\mathfrak {M}}_k (G_d(\cdot ,a,\alpha ,\beta ,\gamma )) \in \mathcal {G}_{d-k}$, its radial part being

$$\begin{aligned}&\frac{\zeta _d(a,\alpha ,\beta ,\gamma )}{\zeta _{d-k}(a,\alpha ,\beta ,\gamma )} g_{d-k}(\cdot ,a,\alpha ,\beta ,\gamma ) \nonumber \\&\quad = \frac{\zeta _d(a,\alpha ,\beta ,\gamma )}{\zeta _{d-k}(a,\alpha ,\beta ,\gamma )} \,g_{d}\left( \cdot ,a,\alpha +\frac{k}{2},\beta +\frac{k}{2},\gamma +\frac{k}{2}\right) . \end{aligned}$$

$\square$

Proof of Theorem 10

The proof follows that of Theorem 9. The condition $(\alpha -\frac{k}{2},\beta -\frac{k}{2},\gamma -\frac{k}{2}) \in \mathcal {P}_{d+k}$ ensures that the downgraded covariance is positive semidefinite in ${\mathbb {R}}^{d+k}$, based on Theorem 1. $\square$

Proof of Theorem 11

The proof relies on expansion (19) of the radial function $r \mapsto g_d(r; a,\alpha ,\beta ,\gamma )$, valid for $r \in [0,a]$ and $\alpha -\frac{d}{2} \not \in {\mathbb {N}}$. Using formulae 5.5.3 and 5.11.12 of Olver et al. (2010), as well as the theorem of dominated convergence to interchange limits and infinite summations, one finds the following asymptotic equivalence:

$$\begin{aligned}&g_d(r; a,\alpha ,\beta ,\gamma ) \sim {}_0F_1\left( ;\frac{d}{2}-\alpha +1;\frac{\beta \gamma r^2}{a^2} \right) \\& +\frac{\varGamma (\frac{d}{2}-\alpha )}{ \varGamma (\alpha -\frac{d}{2})} \left( \frac{\beta \gamma r^2}{a^2}\right) ^{\alpha -\frac{d}{2}} {}_0F_1\left( ;\alpha -\frac{d}{2}+1;\frac{\beta \gamma r^2}{a^2} \right) , \; r \le a, \end{aligned}$$

as $\beta \rightarrow +\infty$ and $\gamma \rightarrow +\infty$ ($f \sim g$ means $f = g(1+\mathcal {O}(1))$, i.e., f and g are asymptotically equivalent). The left-hand side can be expressed in terms of modified Bessel functions of the first ($I_{\eta }$) and second ($K_{\eta }$) kinds thanks to formulae 5.5.3, 10.27.4 and 10.39.9 of Olver et al. (2010), which finally yields:

$$\begin{aligned}g_d(r; a,\alpha ,\beta ,\gamma ) &\sim \varGamma \left( \frac{d}{2}-\alpha +1\right) \left( \frac{\sqrt{\beta \gamma } r}{a}\right) ^{\alpha -\frac{d}{2}} I_{\frac{d}{2}-\alpha }\left( \frac{2\sqrt{\beta \gamma } r}{a}\right) \nonumber \\&\quad +\frac{\varGamma (\frac{d}{2}-\alpha )}{ \varGamma (\alpha -\frac{d}{2})} \varGamma \left( \alpha -\frac{d}{2}+1\right) \left( \frac{\sqrt{\beta \gamma } r}{a}\right) ^{\alpha -\frac{d}{2}} I_{\alpha -\frac{d}{2}}\left( \frac{2\sqrt{\beta \gamma } r}{a}\right) \nonumber \\&\quad = \frac{2}{\varGamma \left( \alpha -\frac{d}{2}\right) } \left( \frac{\sqrt{\beta \gamma } r}{a}\right) ^{\alpha -\frac{d}{2}} K_{\alpha -\frac{d}{2}}\left( \frac{2\sqrt{\beta \gamma } r}{a}\right) , \quad r \le a. \end{aligned}$$

(22)

Accordingly, $g_d(\cdot ; a,\alpha ,\beta ,\gamma )$ tends pointwise to the radial part of the Matérn covariance (12) by letting $\beta$ and $\gamma$ tend to infinity and a be asymptotically equivalent to $2 b \sqrt{\beta \gamma }$. In particular, since a tends to infinity, the pointwise convergence is true for any $r\ge 0$. It is also true if $\alpha -\frac{d}{2} \in {\mathbb {N}}$, as it suffices to consider the asymptotic equivalence (22) with $\alpha -\delta -\frac{d}{2}$ and $\delta >0$ and then to let $\delta$ tend to zero, both the Gauss hypergeometric and Matérn covariances being continuous with respect to the parameter $\alpha$. Note that the conditions of Theorem 1 are fulfilled when $\alpha$ is fixed and greater than $\frac{d}{2}$ and $\beta$ and $\gamma$ become infinitely large, so that $g_d(\cdot ; a,\alpha ,\beta ,\gamma )$ in (22) is the radial part of a valid covariance kernel. Finally, because $g_d(\cdot ; a,\alpha ,\beta ,\gamma )$ is a decreasing function on any compact segment of ${\mathbb {R}}_{\ge 0}$ for sufficiently large a and $\beta$ or $\gamma$ (Theorem 8) and the limit function (the radial part of the Matérn covariance (12)) is continuous on ${\mathbb {R}}_{\ge 0}$, Dini’s second theorem implies that the pointwise convergence is actually uniform on any compact segment of ${\mathbb {R}}_{\ge 0}$. In turn, since all the functions are lower bounded by zero, uniform convergence on a compact segment of ${\mathbb {R}}_{\ge 0}$ implies uniform convergence on ${\mathbb {R}}_{\ge 0}$. $\square$

The proofs of Theorems 12 to 16 use of the same argument as above to identify pointwise convergence with uniform convergence. This argument will be omitted for the sake of brevity.

Proof of Theorem 12

The starting point is the expansion (19) of $g_d(\cdot ; a,\alpha ,\beta ,\gamma )$ in [0, a]. Using formulae 5.5.3 and 5.11.12 of Olver et al. (2010) and the dominated convergence theorem to interchange limits and infinite summations, one finds the following asymptotic equivalence as $\gamma$ tends to infinity:

$$\begin{aligned}&g_d\left( r; a,\alpha ,\beta ,\gamma \right) \sim {}_1F_1\left( \frac{d}{2}-\beta +1;\frac{d}{2}-\alpha +1;-\frac{\gamma \, r^2}{a^2}\right) \nonumber \\&\quad + {}_1F_1\left( \alpha -\beta +1;\alpha -\frac{d}{2}+1;-\frac{\gamma \, r^2}{a^2}\right) \nonumber \\&\quad \frac{\varGamma (\frac{d}{2}-\alpha ) \varGamma (\beta -\frac{d}{2})}{\varGamma (\alpha -\frac{d}{2}) \varGamma (\beta -\alpha )} \left( \frac{\gamma \, r^2}{a^2}\right) ^{\alpha -\frac{d}{2}}, \quad 0 \le r < a. \end{aligned}$$

(23)

Using formula D.8 of Matheron (1965) and letting $a \rightarrow +\infty$ such that $\frac{a}{\sqrt{\gamma }} \rightarrow b > 0$ yields the claim. $\square$

Proof of Theorem 13

The proof relies on (23) and formulae 5.5.3 and 13.2.42 of Olver et al. (2010). $\square$

Proof of Theorem 14

The proof relies on (23) and formulae 8.2.3, 8.2.4, 8.5.1 and 13.6.3 of Olver et al. (2010). $\square$

Proof of Theorem 15

The proof follows from Theorem 11 and the fact that the Matérn covariance (12) with scale parameter $b/(2 \sqrt{\alpha })$ and smoothness parameter $\alpha$ tends to the Gaussian covariance (13) as $\alpha \rightarrow +\infty$. Following Chernih et al. (2014), the convergence can also be shown by noting that the spectral density (8) of the Gauss hypergeometric covariance is asymptotically equivalent to

$$\begin{aligned}&{\widetilde{G}}_d({{\varvec{u}}}; a,\alpha ,\beta ,\gamma ) \sim \left( \frac{\pi a^2 \alpha }{\beta \gamma }\right) ^{\frac{d}{2}} \, \sum _{n=0}^{+\infty } \frac{1}{n!} \left( -\frac{\alpha (\pi a \Vert {{\varvec{u}}}\Vert )^2}{\beta \gamma } \right) ^n \\&\quad = \left( \frac{\pi a^2 \alpha }{\beta \gamma }\right) ^{\frac{d}{2}} \, \exp \left( -\frac{\alpha (\pi a \Vert {{\varvec{u}}}\Vert )^2}{\beta \gamma } \right) , \quad {{\varvec{u}}} \in {\mathbb {R}}^d, \end{aligned}$$

as $\alpha \rightarrow +\infty$, $\beta \rightarrow +\infty$ and $\gamma \rightarrow +\infty$. If, furthermore, $a \rightarrow +\infty$ such that $a \sqrt{\frac{\alpha }{\beta \gamma }} \rightarrow b > 0$, then one obtains:

$$\begin{aligned}&{\widetilde{G}}_d({{\varvec{u}}}; a,\alpha ,\beta ,\gamma ) \sim \pi ^{\frac{d}{2}} b^d \, \exp \left( -(\pi b \Vert {{\varvec{u}}}\Vert )^2 \right) ,\quad {{\varvec{u}}} \in {\mathbb {R}}^d, \end{aligned}$$

which coincides with the spectral density of the Gaussian covariance (13) (Arroyo and Emery 2021; Lantuéjoul 2002). $\square$

Proof of Theorem 16

The proof follows from the asymptotic equivalence (23) for $\gamma$ tending to infinity. As $\beta$ tends to $\alpha$ and a tends to infinity in such a way that $\frac{a}{\sqrt{\gamma }}$ tends to $b>0$, the first term in the right-hand side of (23) tends to $\exp (-r^2/b^2)$ and the second term to zero. $\square$

Proof of Lemma 3

One has $\Phi _1({{\varvec{x}}}) = \varphi _1 \circ \psi _1(\Vert {\varvec{x}}\Vert )$, where $\varphi _1: x \mapsto {}_1 F_2\left( \alpha ;\beta ,\gamma ; -x\right)$ is an infinitely differentiable function on ${\mathbb {R}}_{\ge 0}$, with (Olver et al. 2010, formula 16.3.1)

$$\begin{aligned}&(-1)^k \frac{\partial ^k \varphi _1}{\partial x^k}(x) = \frac{\varGamma (\alpha +k) \varGamma (\beta ) \varGamma (\gamma )}{\varGamma (\alpha ) \varGamma (\beta +k) \varGamma (\gamma +k)}\\&\quad \times {}_1 F_2\left( \alpha +k;\beta +k,\gamma +k; -x\right) , \quad x \in {\mathbb {R}}_{\ge 0}, k \in {\mathbb {N}}. \end{aligned}$$

If $(\alpha +q+2,\beta +q+2,\gamma +q+2) \in \mathcal {P}_0$, then, for any $k=0, \ldots , q+2$, $(\alpha +k,\beta +k,\gamma +k) \in \mathcal {P}_0$ and $(-1)^k \frac{\partial ^k \varphi _1}{\partial x^k}$ is nonnegative on ${\mathbb {R}}_{\ge 0}$, hence $\varphi _1$ is $(q+2)$-times monotone. Since $\psi _1$ is positive and has a $(q+1)$-times monotone derivative, the composite function $\varphi _1 \circ \psi _1$ is $(q+2)$-times monotone (Gneiting 1999, proposition 4.5). The fact that this composite function is continuous implies that $\Phi _1$ is positive semidefinite in ${\mathbb {R}}^{2q+1}$ (Askey 1973; Micchelli 1986; Gneiting 1999, criterion 1.3). $\square$

Proof of Lemma 4

$\Phi _2({{\varvec{y}}}) = \varphi _2 \circ \psi _2(\Vert {\varvec{y}}\Vert )$, where $\varphi _2: \alpha \mapsto {}_1 F_2\left( \alpha ;\alpha +1,\gamma ; -x\right)$ is a nonnegative function on $[0,\alpha _{\max }]$, insofar as $(\alpha ,\alpha +1,\gamma ) \in \mathcal {P}_0$ as soon as $\alpha \le \alpha _{\max }$. This function is infinitely differentiable; for $k \in {\mathbb {N}}_{>0}$, its k-th derivative, obtained with a term-by-term differentiation of (1), is

$$\begin{aligned}&\frac{\partial ^k \varphi _2}{\partial \alpha ^k}(\alpha )\nonumber \\&\quad = \sum _{n=1}^{+\infty } \frac{(-1)^{k-1} k! \, n \, \varGamma (\gamma )}{n! (\alpha +n)^{k+1} \varGamma (\gamma +n)} \left( -x\right) ^n \nonumber \\&\quad = \frac{(-1)^{k} k! \, x}{(\alpha +1)^{k+1} \gamma } \sum _{n=0}^{+\infty } \frac{(\alpha +1)^{k+1} \varGamma (\gamma +1)}{n! (\alpha +1+n)^{k+1} \varGamma (\gamma +1+n)} \left( -x\right) ^n \nonumber \\&\quad = \frac{(-1)^{k} k! \, x}{(\alpha +1)^{k+1} \gamma } \, {}_{k+1}F_{k+2}\left( \alpha +1,\ldots ,\alpha +1;\alpha +2,\ldots ,\alpha +2,\gamma +1;-x\right) . \end{aligned}$$

(24)

If $\alpha \in [0,\alpha _{\max }]$, then $(\alpha +1,\alpha +2,\gamma +1) \in \mathcal {P}_0$ and ${}_{k+1}F_{k+2}(\alpha +1,\ldots ,\alpha +1;\alpha +2,\ldots ,\alpha +2,\gamma +1;-x)$ is nonnegative, as a beta mixture of nonnegative ${}_{1}F_{2}$ functions (Olver et al. 2010, formula 16.5.2), which implies that $\varphi _2$ is completely monotone on $[0,\alpha _{\max }]$. Since $\psi _2$ is positive with values in $[0,\alpha _{\max }]$ and has a $(q^{\prime }+1)$-times monotone derivative, the composition $\varphi _2 \circ \psi _2$ is $(q^{\prime }+2)$-times monotone on ${\mathbb {R}}_{\ge 0}$. As it is continuous, this entails that $\Phi _2$ is a positive semidefinite in ${\mathbb {R}}^{2q^{\prime }+1}$ (Micchelli 1986). $\square$

Proof of Lemma 5

Let $\Phi ({\varvec{x}},{\varvec{y}}) = \varphi (\psi _1(\Vert {\varvec{x}}\Vert ),\psi _2(\Vert {\varvec{y}}\Vert ))$, where $\varphi : (x,\alpha ) \mapsto \frac{1}{x} \, {}_1 F_2\left( \alpha ;\alpha +1,\gamma ; -x\right)$ is nonnegative and infinitely differentiable on ${\mathbb {R}}_{>0} \times [0,\alpha _{\max }]$. From (24), it comes, for $k, k^\prime \in {\mathbb {N}}$:

$$\begin{aligned}&\frac{\partial ^{k+k^{\prime }} \varphi }{\partial x^k \, \partial \alpha ^{k^{\prime }}}(x,\alpha ) \\&\quad = \sum _{n=0}^{+\infty } \frac{(-1)^{k+k^{\prime }} k^{\prime }! \, \varGamma (\gamma )}{\varGamma (n+1) (\alpha +n+k+1)^{k^{\prime }+1} \varGamma (\gamma +n+k+1)} \left( -x\right) ^{n} \\&\quad = \frac{(-1)^{k+k^{\prime }} k^{\prime }! \, \varGamma (\gamma )}{(\alpha +k+1)^{k^{\prime }+1} \varGamma (\gamma +k+1)} \\&\qquad \times {}_{k^{\prime }+1}F_{k^{\prime }+2}\left( \alpha +k+1,\ldots ,\alpha +k+1; \right.\\&\qquad \qquad \qquad \quad \; \; \left.\alpha +k+2,\ldots ,\alpha +k+2,\gamma +k+1;-x \right) . \end{aligned}$$

If $(\alpha +q+3,\alpha +q+4,\gamma +q+3) \in \mathcal {P}_0$, then, for any $k=0, \ldots , q+2$, $(\alpha +k+1,\alpha +k+2,\gamma +k+1) \in \mathcal {P}_0$ and the hypergeometric term ${}_{k^{\prime }+1}F_{k^{\prime }+2}$ is nonnegative, as a beta mixture of nonnegative ${}_{1}F_{2}$ terms. Under this condition, $(-1)^{k+k^{\prime }} \frac{\partial ^{k+k^{\prime }} \varphi }{\partial x^k \, \partial \alpha ^{k^{\prime }}}$ is nonnegative for $k=0, \ldots , q+2$ and any $k^\prime \in {\mathbb {N}}$. Accordingly, $\varphi$ is a bivariate multiply monotone function of order $(q+2,q^\prime +2)$, and so is the composite function $\varphi (\psi _1,\psi _2)$ (Gneiting 1999, proposition 4.5). Arguments in Williamson (1956) generalized to functions of two variables imply that $\varphi (\psi _1,\psi _2)$ is a mixture of products of truncated power functions of the form (15) (one function of x with power exponent $q+1$ times one function of $\alpha$ with power exponent $q^\prime +1$) and is the radial part of a product covariance kernel in ${\mathbb {R}}^{2q+1} \times {\mathbb {R}}^{2q^{\prime }+1}$. $\square$

Proof of Theorem 17

We start proving (1). Conditions (i), (ii) and (v) imply the existence of a spectral density associated with each direct or cross covariance (Theorem 1). Based on Cramér’s criterion (Cramér 1940; Chilès and Delfiner 2012), $\varvec{{\widetilde{G}}}_d(\cdot ; a{\varvec{1}},\alpha {\varvec{1}},\varvec{\beta },\varvec{\gamma },\varvec{\rho })$ is a valid matrix-valued spectral density function if, and only if, $\varvec{{\widetilde{G}}}_d({\varvec{u}}; a {\varvec{1}},\alpha {\varvec{1}},\varvec{\beta },\varvec{\gamma },\varvec{\rho })$ is positive semidefinite for any vector ${\varvec{u}} \in {\mathbb {R}}^d$. The key of the proof is to expand this matrix as a positive mixture of positive semidefinite matrices. Such an expansion rests on the following identity, which can be obtained by a term-by-term integration of the infinite series (1) defining the generalized hypergeometric function ${}_1F_2$ along with formula 3.251.1 of Gradshteyn and Ryzhik (2007):

$$\begin{aligned}&\int _0^1 \int _0^1 {}_1F_2 \left( \alpha ;\beta ,\gamma ;-t_1 t_2 (a\,x)^2\right) \nonumber \\&\qquad t_1^{\beta -1}(1-t_1)^{\beta _{ij}-\beta -1} t_2^{\gamma -1}(1-t_2)^{\gamma _{ij}-\gamma -1} \text {d}t_1 \text {d}t_2 \nonumber \\&\quad = \frac{\varGamma (\beta ) \varGamma (\beta _{ij}-\beta ) \varGamma (\gamma ) \varGamma (\gamma _{ij}-\gamma )}{\varGamma (\beta _{ij}) \varGamma (\gamma _{ij})}\nonumber \\&\qquad \times {}_1F_2\left( \alpha ;\beta _{ij},\gamma _{ij};-(a\,x)^2\right) , \end{aligned}$$

(25)

for $x \ge 0, a>0, \alpha>0, \beta _{ij}>\beta >0$ and $\gamma _{ij}>\gamma >0$. Accordingly, for ${\varvec{u}} \in {\mathbb {R}}^d$:

$$\begin{aligned}&{\widetilde{{\varvec{G}}}}_d({\varvec{u}}; {a}{\varvec{1}},{\alpha }{\varvec{1}},\varvec{\beta },\varvec{\gamma },\varvec{\rho })\\&\quad = \frac{\pi ^{\frac{d}{2}} a^{d} \varGamma (\alpha ) \varGamma (\varvec{\beta } - \frac{d}{2}) \varGamma (\varvec{\gamma } - \frac{d}{2}) \varvec{\rho }}{\varGamma (\alpha -\frac{d}{2}) \varGamma (\beta ) \varGamma (\varvec{\beta }-\beta ) \varGamma (\gamma ) \varGamma (\varvec{\gamma }-\gamma )} \\&\times \int _0^1 \int _0^1 {}_1F_2\left( \alpha ;\beta ,\gamma ;-{t_1 \, t_2 (\pi a \Vert {\varvec{u}}\Vert )^2}\right) \\&\quad t_1^{\beta -1}(1-t_1)^{\varvec{\beta }-\beta -1} t_2^{\gamma -1}(1-t_2)^{\varvec{\gamma }-\gamma -1} \text {d}t_1 \text {d}t_2, \end{aligned}$$

with the products, quotients and powers taken element-wise. ${}_1F_2\left( \alpha ;\beta ,\gamma ;-{t_1 t_2 (\pi a \Vert {\varvec{u}}\Vert )^2}\right)$ is nonnegative for any $t_1, t_2 \in [0,1]$ under Condition (vi) (Cho et al. 2020). Under Conditions (iii)–(iv), $(1-t_1)^{\varvec{\beta }}$ and $(1-t_2)^{\varvec{\gamma }}$ are positive semidefinite matrices (Lemma 1). Along with Condition (vii), ${\widetilde{{\varvec{G}}}}_d({\varvec{u}}; a {\varvec{1}},\alpha {\varvec{1}},\varvec{\beta },\varvec{\gamma },\varvec{\rho })$ is positive semidefinite for any ${\varvec{u}}$ in ${\mathbb {R}}^d$, as the elewent-wise product of positive semidefinite matrices, which completes the proof for (1).

We now prove (2). Under Condition (vi), the generalized hypergeometric function ${}_1F_2(\alpha ;\beta ,\gamma ,x)$ is positive and increasing in x on ${\mathbb {R}}$ (Olver et al. 2010, formula 16.3.1). Therefore, if ${\varvec{a}}$ fulfills Condition (i), $[{}_1F_2\left( \alpha ;\beta ,\gamma ;-t_1 t_2(\,a_{ij}\, x)^2\right) ]_{i,j=1}^p$ is positive semidefinite, as the sum of a min matrix with positive entries (Horn and Johnson 2013, problem 7.1.P18) and a diagonal matrix with nonnegative entries. The proof of (1) can then be adapted in a straightforward manner, by substituting such a positive semidefinite matrix for the positive scalar ${}_1F_2\left( \alpha ;\beta ,\gamma ;-t_1 t_2(\,a\, x)^2\right)$.

The proof of (3) follows that of (2) and relies on the fact that, under Conditions (i) and (vi), the matrix $[{}_1F_2\left( \alpha ;\beta ,\gamma ;-t_1 t_2 (\,a_{ij}\, x)^2\right) ]_{i,j=1}^p$ is positive semidefinite for any $t_1$, $t_2$ and x (Lemma 3).

The proof of (4) is similar to that of (1), with (25) replaced by

$$\begin{aligned}&\int _0^1 \int _0^1 {}_1F_2 \left( \alpha _{ij};\alpha _{ij}+1,\gamma ;-t_1 t_2 (a_{ij}\,x)^2\right) \\&\qquad t_1^{\alpha _{ij}}(1-t_1)^{\beta _{ij}-\alpha _{ij}-2} t_2^{\gamma -1}(1-t_2)^{\gamma _{ij}-\gamma -1} \text {d}t_1 \text {d}t_2\\&\quad = \frac{\varGamma (\alpha _{ij}+1) \varGamma (\beta _{ij}-\alpha _{ij}-1) \varGamma (\gamma ) \varGamma (\gamma _{ij}-\gamma )}{\varGamma (\beta _{ij}) \varGamma (\gamma _{ij})}\\&\qquad \times {}_1F_2\left( \alpha _{ij};\beta _{ij},\gamma _{ij};-(a_{ij}\,x)^2\right) , \end{aligned}$$

for $x \ge 0, a_{ij}>0, \beta _{ij}-1>\alpha _{ij}>0$ and $\gamma _{ij}>\gamma >0$ for i, j in $[1,\ldots ,p]$. Under Condition (ii), the composite function $t \mapsto \exp (-x(\psi _2(t)-\psi _2(0)))$ is $(q^{\prime }+2)$-times monotone (Gneiting 1999, proposition 4.5), hence it is a mixture of truncated power functions of the form (15) and is the radial part of a positive semidefinite function in ${\mathbb {R}}^{2q^{\prime }+1}$ for any $x>0$. A classical result by Schoenberg (1938) states that ${\varvec{x}} \mapsto \psi _2(\Vert {\varvec{x}}\Vert )-\psi _2(0)$ is a variogram in ${\mathbb {R}}^{2q^{\prime }+1}$, so $\varvec{\alpha }$ is conditionally negative semidefinite (Example 1) and $[t_1^{\alpha _{ij}}]_{i,j=1}^p$ is positive semidefinite for any $t_1 \in [0,1]$ (Lemma 1). Under Conditions (iii) and (iv), $[(1-t_1)^{\beta _{ij}-\alpha _{ij}}]_{i,j=1}^p$ and $[(1-t_2)^{\gamma _{ij}}]_{i,j=1}^p$ are positive semidefinite for any $t_1, t_2 \in [0,1]$ (Lemma 1). Under Condition (ii), $[{}_1F_2 (\alpha _{ij};\alpha _{ij}+1,\gamma ;-t_1 t_2 (a\,x)^2)]_{i,j=1}^p$ is also positive semidefinite for any $t_1, t_2 \in [0,1]$, $a>0$, $x>0$ (Lemma 4). As $(\alpha _{ij}+1,\alpha _{ij}+2,\gamma +1) \in \mathcal {P}_0$, the generic entry of this matrix decreases with a (Olver et al. 2010, formula 16.3.1), hence the matrix $[{}_1F_2 (\alpha _{ij};\alpha _{ij}+1,\gamma ;-t_1 t_2 (a_{ij}\,x)^2)]_{i,j=1}^p$ has increased diagonal entries and is still positive semidefinite. Finally, Condition (v) and the Schur’s product theorem imply that ${\widetilde{{\varvec{G}}}}_d({\varvec{u}}; {\varvec{a}},\varvec{\alpha },\varvec{\beta },\varvec{\gamma },\varvec{\rho })$ is positive semidefinite for any ${\varvec{u}}$ in ${\mathbb {R}}^d$, as the elewent-wise product of positive semidefinite matrices, which completes the proof of (4).

The proof of (5) follows the same line of reasoning as that of (4). The positive semidefiniteness of $[a_{ij}^{-2} \, {}_1F_2 (\alpha _{ij};\alpha _{ij}+1,\gamma ;-t_1 t_2 (a_{ij}\,x)^2)]_{i,j=1}^p$ now stems from Conditions (i), (ii) and (v) together with Lemma 5.

$\square$