2. Conditional negative definiteness for matrix-valued functions

Real-valued conditionally negative definite functions are essential to characterizing univariate variograms. A function $\gamma\,:\, {{\mathbb{R}}}^d \rightarrow {{\mathbb{R}}}$ is a univariate variogram if and only if $\gamma(0)=0$ and $\gamma$ is conditionally negative definite, that is, $\gamma$ is symmetric and for all $n \ge 2$ , $\boldsymbol{x}_{\textbf{1}},\ldots,\boldsymbol{x}_{\boldsymbol{n}} \in {{\mathbb{R}}}^d$ , $a_1,\ldots,a_n \in {{\mathbb{R}}}$ such that $\sum_{k=1}^n a_k=0$ , the inequality $\sum_{i=1}^n\sum_{j=1}^n a_i \gamma(\boldsymbol{x}_{\boldsymbol{i}}-\boldsymbol{x}_{\boldsymbol{j}})a_j \le 0$ holds [Reference Matheron15]. An extended notion of conditional negative definiteness for matrix-valued functions is part of a characterization of cross-variograms. A function ${\boldsymbol {\tilde\gamma}}\,:\, {{\mathbb{R}}}^d \rightarrow {{\mathbb{R}}}^{m \times m}$ is a cross-variogram if and only if ${\boldsymbol {\tilde\gamma}}(\textbf{0}) =\textbf{0}$ , ${\boldsymbol {\tilde\gamma}}(\boldsymbol{h})={\boldsymbol {\tilde\gamma}}({-}\boldsymbol{h})={\boldsymbol {\tilde\gamma}}(\boldsymbol{h})^\top$ and

(1) \begin{equation} \sum_{i=1}^n\sum_{j=1}^n \boldsymbol{a}_{\boldsymbol{i}}^\top {\boldsymbol {\tilde\gamma}}(\boldsymbol{x}_{\boldsymbol{i}}-\boldsymbol{x}_{\boldsymbol{j}})\boldsymbol{a}_{\boldsymbol{j}} \le 0\end{equation}

for $n \ge 2$ , $\boldsymbol{x}_{\textbf{1}},\ldots,\boldsymbol{x}_{\boldsymbol{n}} \in {{\mathbb{R}}}^d$ , $\boldsymbol{a}_{\textbf{1}},\ldots,\boldsymbol{a}_{\boldsymbol{n}} \in {{\mathbb{R}}}^m$ such that $\sum_{k=1}^n \boldsymbol{a}_{\boldsymbol{k}} =\textbf{0}$ [Reference Ma14]. A function satisfying condition (1) is called an almost negative definite matrix-valued function in [Reference Xie31, p. 40].

A pseudo cross-variogram ${\boldsymbol \gamma}$ has similar, but only necessary properties; see [Reference Du and Ma7]. It holds that $\gamma_{ii}(\textbf{0})=0$ , and $\gamma_{ij}(\boldsymbol{h})=\gamma_{ji}({-}\boldsymbol{h})$ , $i,j=1,\ldots,m$ . Additionally, a pseudo cross-variogram is an almost negative definite matrix-valued function as well, but inequality (1), loosely speaking, cannot enforce non-negativity on the secondary diagonals. Therefore we consider the following stronger notion of conditional negative definiteness; see [Reference Gesztesy and Pang9].

Definition 1. A function ${\boldsymbol \gamma}\,:\, {{\mathbb{R}}}^d \rightarrow {{\mathbb{R}}}^{m \times m}$ is called conditionally negative definite if

(2a) \begin{align}\gamma_{ij}(\boldsymbol{h})=\gamma_{ji}({-}\boldsymbol{h}), \quad i,j=1,\ldots,m,\end{align}

(2b) \begin{align}\sum_{i=1}^n\sum_{j=1}^n \boldsymbol{a}_{\boldsymbol{i}}^\top {\boldsymbol \gamma}(\boldsymbol{x}_{\boldsymbol{i}}-\boldsymbol{x}_{\boldsymbol{j}})\boldsymbol{a}_{\boldsymbol{j}} \le 0, \end{align}

for all $n \in \mathbb{N}$ , $\boldsymbol{x}_{\textbf{1}},\ldots,\boldsymbol{x}_{\boldsymbol{n}} \in {{\mathbb{R}}}^d$ , $\boldsymbol{a}_{\textbf{1}},\ldots,\boldsymbol{a}_{\boldsymbol{n}} \in {{\mathbb{R}}}^m$ such that

\[\textbf{1}_{\boldsymbol{m}}^\top\sum_{k=1}^n \boldsymbol{a}_{\boldsymbol{k}} =0\]

with $\textbf{1}_{\boldsymbol{m}}\,:\!=\, (1,\ldots, 1)^\top \in {{\mathbb{R}}}^m$ .

Obviously, the set of conditionally negative definite matrix-valued functions is a convex cone which is closed under integration and pointwise limits, if existing. In the univariate case, the concepts of conditionally and almost negative definite functions coincide, reproducing the traditional notion of real-valued conditionally negative definite functions. The main difference between them is the broader spectrum of vectors for which inequality (2b) has to hold, in that the sum of all components has to be zero instead of each component of the sum itself. This modification in particular includes sets of linearly independent vectors in the pool of admissible test vector families, resulting in more restrictive conditions on the secondary diagonals. Indeed, choosing $n=2$ , $\boldsymbol{x}_{\textbf{1}}=\boldsymbol{h} \in {{\mathbb{R}}}^d$ , $\boldsymbol{x}_{\textbf{2}}=\textbf{0}$ , and $\boldsymbol{a}_{\textbf{1}}=\boldsymbol{e}_{\boldsymbol{i}}$ , $\boldsymbol{a}_{\textbf{2}}=-\boldsymbol{e}_{\boldsymbol{j}}$ in Definition 1 with $\{\boldsymbol{e}_{\textbf{1}},\ldots,\boldsymbol{e}_{\boldsymbol{m}}\}$ denoting the canonical basis in ${{\mathbb{R}}}^m$ , we have $\gamma_{ij}(\boldsymbol{h}) \ge 0$ for a conditionally negative definite function ${\boldsymbol \gamma}\,:\, {{\mathbb{R}}}^d \rightarrow {{\mathbb{R}}}^{m \times m}$ with $\gamma_{ii}(\textbf{0}) = 0$ , $i,j=1,\ldots,m$ , fitting the non-negativity of a pseudo cross-variogram. In fact, the latter condition on the main diagonal and the conditional negative definiteness property are sufficient to characterize pseudo cross-variograms.

Proof. The proof is analogous to the univariate one in [Reference Matheron15]. Let $\boldsymbol{Z}$ be an m-variate random field with pseudo cross-variogram ${\boldsymbol \gamma}$ . Obviously, $\gamma_{ii}(\textbf{0}) = 0$ and $\gamma_{ij}(\boldsymbol{h})=\gamma_{ji}({-}\boldsymbol{h})$ for all $\boldsymbol{h} \in {{\mathbb{R}}}^d$ , $i,j=1,\ldots, m$ . Define an m-variate random field $\tilde{\boldsymbol{Z}}$ via $\tilde Z_i(\boldsymbol{x})=Z_i(\boldsymbol{x})-Z_1(\textbf{0})$ , $\boldsymbol{x} \in {{\mathbb{R}}}^d$ , $i=1,\ldots,m$ . Then $\boldsymbol{Z}$ and $\tilde{\boldsymbol{Z}}$ have the same pseudo cross-variogram, and

\[{{{ Cov}}}(\tilde Z_i(\boldsymbol{x}),\tilde Z_j(\boldsymbol{y})) = \gamma_{i1}(\boldsymbol{x})+\gamma_{j1}(\boldsymbol{y})-\gamma_{ij}(\boldsymbol{x}-\boldsymbol{y})\]

(see also [Reference Papritz, Künsch and Webster22, equation (6)]), that is,

\[{{{ Cov}}}(\tilde{\boldsymbol{Z}}(\boldsymbol{x}), \tilde{\boldsymbol{Z}}(\boldsymbol{y})) = \boldsymbol{\gamma}_{\textbf{1}}(\boldsymbol{x})\textbf{1}_{\boldsymbol{m}}^\top + \textbf{1}_{\boldsymbol{m}}\boldsymbol{\gamma}_{\textbf{1}}^\top(\boldsymbol{y}) - {\boldsymbol \gamma}(\boldsymbol{x} - \boldsymbol{y}), \quad \boldsymbol{x}, \boldsymbol{y} \in {{\mathbb{R}}}^d,\]

with $\boldsymbol{\gamma}_{\textbf{1}}(\boldsymbol{x})\,:\!=\, (\gamma_{11}(\boldsymbol{x}),\ldots,\gamma_{m1}(\boldsymbol{x}))^\top$ . For $\textbf{1}_{\boldsymbol{m}}^\top \sum_{k=1}^m \boldsymbol{a}_{\boldsymbol{k}}=0$ , we thus have

\begin{align*} 0 & \le {Var}\Biggl(\sum_{i=1}^n \boldsymbol{a}_{\boldsymbol{i}}^\top\tilde{\boldsymbol{Z}}(\boldsymbol{x}_{\boldsymbol{i}})\Biggr)\\ &=\sum_{i=1}^n \sum_{j=1}^n \boldsymbol{a}_{\boldsymbol{i}}^\top \bigl(\boldsymbol{\gamma}_{\textbf{1}}(\boldsymbol{x}_{\boldsymbol{i}})\textbf{1}_{\boldsymbol{m}}^\top +\textbf{1}_{\boldsymbol{m}}\boldsymbol{\gamma}_{\textbf{1}}^\top(\boldsymbol{x}_{\boldsymbol{j}})- {\boldsymbol \gamma}(\boldsymbol{x}_{\boldsymbol{i}} -\boldsymbol{x}_{\boldsymbol{j}})\bigr) \boldsymbol{a}_{\boldsymbol{j}}\\& = -\sum_{i=1}^n \sum_{j=1}^n \boldsymbol{a}_{\boldsymbol{i}}^\top {\boldsymbol \gamma}(\boldsymbol{x}_{\boldsymbol{i}} -\boldsymbol{x}_{\boldsymbol{j}}) \boldsymbol{a}_{\boldsymbol{j}}.\end{align*}

Now let ${\boldsymbol \gamma}$ be conditionally negative definite and $\gamma_{ii}(\textbf{0})=0$ , $i=1,\ldots,m$ . Let $\boldsymbol{a}_{\textbf{1}},\ldots, \boldsymbol{a}_{\boldsymbol{n}} \in {{\mathbb{R}}}^m, \boldsymbol{x}_{\textbf{1}}, \ldots,\boldsymbol{x}_{\boldsymbol{n}} \in {{\mathbb{R}}}^d$ be arbitrary, $\boldsymbol{x}_{\textbf{0}}=\textbf{0} \in {{\mathbb{R}}}^d$ and

\[\boldsymbol{a}_{\textbf{0}} = \Biggl({-}\textbf{1}_{\boldsymbol{m}}^\top\sum_{k=1}^n \boldsymbol{a}_{\boldsymbol{k}}, 0, \ldots,0\Biggr)\in {{\mathbb{R}}}^{m}.\]

Then

\begin{align*}0 &\le - \sum_{i=0}^n \sum_{j=0}^n \boldsymbol{a}_{\boldsymbol{i}}^\top {\boldsymbol \gamma}(\boldsymbol{x}_{\boldsymbol{i}}-\boldsymbol{x}_{\boldsymbol{j}}) \boldsymbol{a}_{\boldsymbol{j}}\\&= - \sum_{i=1}^n \sum_{j=1}^n \boldsymbol{a}_{\boldsymbol{i}}^\top {\boldsymbol \gamma}(\boldsymbol{x}_{\boldsymbol{i}} -\boldsymbol{x}_{\boldsymbol{j}}) \boldsymbol{a}_{\boldsymbol{j}}- \sum_{i=1}^n \boldsymbol{a}_{\boldsymbol{i}}^\top{\boldsymbol \gamma}(\boldsymbol{x}_{\boldsymbol{i}} -\boldsymbol{x}_{\textbf{0}}) \boldsymbol{a}_{\textbf{0}}- \sum_{j=1}^n \boldsymbol{a}_{\textbf{0}}^\top {\boldsymbol \gamma}(\boldsymbol{x}_{\textbf{0}} -\boldsymbol{x}_{\boldsymbol{j}}) \boldsymbol{a}_{\boldsymbol{j}} \\& \quad- \boldsymbol{a}_{\textbf{0}}^\top {\boldsymbol \gamma}(\boldsymbol{x}_{\textbf{0}} -\boldsymbol{x}_{\textbf{0}}) \boldsymbol{a}_{\textbf{0}}.\end{align*}

Since $\gamma_{11}(\textbf{0})=0$ , and $\boldsymbol{a}_{\textbf{0}}^\top {\boldsymbol \gamma}(\boldsymbol{x}_{\textbf{0}} -\boldsymbol{x}_{\boldsymbol{j}}) \boldsymbol{a}_{\boldsymbol{j}} =\boldsymbol{a}_{\boldsymbol{j}}^\top {\boldsymbol \gamma}(\boldsymbol{x}_{\boldsymbol{j}}) \boldsymbol{a}_{\textbf{0}}$ due to property (2a), we get that

\begin{align*}0 &\le - \sum_{i=0}^n \sum_{j=0}^n \boldsymbol{a}_{\boldsymbol{i}}^\top {\boldsymbol \gamma}(\boldsymbol{x}_{\boldsymbol{i}} -\boldsymbol{x}_{\boldsymbol{j}}) \boldsymbol{a}_{\boldsymbol{j}} \\&= \sum_{i=1}^n \sum_{j=1}^n \boldsymbol{a}_{\boldsymbol{i}}^\top\bigl(\boldsymbol{\gamma}_{\textbf{1}}(\boldsymbol{x}_{\boldsymbol{i}})\textbf{1}_{\boldsymbol{m}}^\top + \textbf{1}_{\boldsymbol{m}}\boldsymbol{\gamma}_{\textbf{1}}^\top(\boldsymbol{x}_{\boldsymbol{j}}) -{\boldsymbol \gamma}(\boldsymbol{x}_{\boldsymbol{i}} -\boldsymbol{x}_{\boldsymbol{j}})\bigr) \boldsymbol{a}_{\boldsymbol{j}},\end{align*}

that is,

\[(\boldsymbol{x},\boldsymbol{y}) \mapsto \boldsymbol{\gamma}_{\textbf{1}}(\boldsymbol{x})\textbf{1}_{\boldsymbol{m}}^\top + \textbf{1}_{\boldsymbol{m}}\boldsymbol{\gamma}_{\textbf{1}}^\top(\boldsymbol{y}) -{\boldsymbol \gamma}(\boldsymbol{x} -\boldsymbol{y})\]

is a matrix-valued positive definite kernel. Let

\[\{\boldsymbol{Z}(\boldsymbol{x})=(Z_1(\boldsymbol{x}),\ldots, Z_m(\boldsymbol{x}))^\top, \boldsymbol{x} \in {{\mathbb{R}}}^d\}\]

be a corresponding centred Gaussian random field. We have to show that ${Var}(Z_i(\boldsymbol{x}+\boldsymbol{h}) - Z_j(\boldsymbol{x}))$ is independent of $\boldsymbol{x}$ for all $\boldsymbol{x}, \boldsymbol{h} \in {{\mathbb{R}}}^d$ , $i,j=1,\ldots,m$ . We even show that $\boldsymbol{x} \mapsto Z_i(\boldsymbol{x}+\boldsymbol{h}) - Z_j(\boldsymbol{x})$ is weakly stationary for $i,j=1,\ldots,m$ :

\begin{align*}& {{{ Cov}}}(Z_i(\boldsymbol{x}+\boldsymbol{h}) - Z_j(\boldsymbol{x}), Z_i(\boldsymbol{y}+\boldsymbol{h}) - Z_j(\boldsymbol{y}))\\&\quad =\gamma_{i1}(\boldsymbol{x}+\boldsymbol{h}) + \gamma_{i1}(\boldsymbol{y}+\boldsymbol{h}) -\gamma_{ii}(\boldsymbol{x}-\boldsymbol{y}) +\gamma_{j1}(\boldsymbol{x}) + \gamma_{j1}(\boldsymbol{y}) - \gamma_{jj}(\boldsymbol{x}-\boldsymbol{y})\\&\quad\quad-\gamma_{j1}(\boldsymbol{x}) - \gamma_{i1}(\boldsymbol{y}+\boldsymbol{h}) + \gamma_{ji}(\boldsymbol{x}-\boldsymbol{y}-\boldsymbol{h}) -\gamma_{i1}(\boldsymbol{x}+\boldsymbol{h}) - \gamma_{j1}(\boldsymbol{y}) +\gamma_{ij}(\boldsymbol{x}+\boldsymbol{h}-\boldsymbol{y})\\&\quad = -\gamma_{ii}(\boldsymbol{x}-\boldsymbol{y})- \gamma_{jj}(\boldsymbol{x}-\boldsymbol{y})+ \gamma_{ji}(\boldsymbol{x}-\boldsymbol{y}-\boldsymbol{h})+\gamma_{ij}(\boldsymbol{x}-\boldsymbol{y}+\boldsymbol{h}).\end{align*}

Theorem 1 answers the questions raised in [Reference Du and Ma7, p. 422] and also settles a question in [Reference Ma13, p. 239] in a more general framework with regard to the intersection of the sets of pseudo and cross-variograms. It turns out that this intersection is trivial in the following sense.

Corollary 1. Let

\begin{align*} \mathcal{P}&=\{{\boldsymbol \gamma}\,:\, {{\mathbb{R}}}^d \rightarrow {{\mathbb{R}}}^{m \times m}\mid {\boldsymbol \gamma}\ {pseudo\ cross\mbox{-}variogram} \},\\ \mathcal{C}&=\{{\boldsymbol {\tilde\gamma}}\,:\, {{\mathbb{R}}}^d \rightarrow {{\mathbb{R}}}^{m \times m}\mid {\boldsymbol {\tilde\gamma}}\ {cross\mbox{-}variogram} \}. \end{align*}

Then we have

(3) \begin{equation} \mathcal{P} \cap \mathcal{C} = \{\textbf{1}_{\boldsymbol{m}}\textbf{1}_{\boldsymbol{m}}^\top \gamma \mid \gamma\,:\, {{\mathbb{R}}}^d \rightarrow {{\mathbb{R}}}\ {variogram} \}. \end{equation}

Proof. Let ${\boldsymbol \gamma} \in \mathcal{P} \cap \mathcal{C}$ . Without loss of generality, assume $m=2$ . Since ${\boldsymbol \gamma} \in \mathcal{P}\cap \mathcal{C}$ , for $n=2$ , $\boldsymbol{x}_{\textbf{1}}=\boldsymbol{h}$ , $\boldsymbol{x}_{\textbf{2}}=\textbf{0}$ , $\boldsymbol{a}_{\textbf{1}},\boldsymbol{a}_{\textbf{2}} \in {{\mathbb{R}}}^2$ with $\textbf{1}_{\textbf{2}}^\top \sum_{k=1}^2 \boldsymbol{a}_{\boldsymbol{k}}=0$ , using the symmetry of ${\boldsymbol \gamma}$ and ${\boldsymbol \gamma}(\textbf{0})=\textbf{0}$ we have

\begin{align*} 0 &\ge \sum_{i=1}^2\sum_{j=1}^2 \boldsymbol{a}_{\boldsymbol{i}}^\top {\boldsymbol \gamma}(\boldsymbol{x}_{\boldsymbol{i}}-\boldsymbol{x}_{\boldsymbol{j}})\boldsymbol{a}_{\boldsymbol{j}} \\ &= 2a_{11}a_{21} \gamma_{11}(\boldsymbol{h})+2a_{12}a_{22}\gamma_{22}(\boldsymbol{h})+2(a_{11}a_{22}+a_{12}a_{21})\gamma_{12}(\boldsymbol{h}) .\end{align*}

Choosing $\boldsymbol{a}_{\textbf{1}}=({-}1,0)^\top$ , $\boldsymbol{a}_{\textbf{2}}= (1-k,k)^\top$ , $k \ge 2$ , and applying the Cauchy–Schwarz inequality due to ${\boldsymbol \gamma} \in \mathcal{C}$ gives

(4) \begin{align}0 \le \gamma_{11}(\boldsymbol{h}) \le \dfrac{-k}{1-k}\gamma_{12}(\boldsymbol{h}) \le \dfrac{1}{1-{{1}/{k}}}\sqrt{\gamma_{11}(\boldsymbol{h})}\sqrt{\gamma_{22}(\boldsymbol{h})}.\end{align}

By symmetry, we also have

(5) \begin{align}0 \le \gamma_{22}(\boldsymbol{h}) &\le \dfrac{ -k}{1-k}\gamma_{12}(\boldsymbol{h}) \le \dfrac{1}{1-{{1}/{k}}}\sqrt{\gamma_{11}(\boldsymbol{h})}\sqrt{\gamma_{22}(\boldsymbol{h})}.\end{align}

Assume first that, without loss of generality, $\gamma_{11}(\boldsymbol{h})=0$ . Then $\gamma_{12}(\boldsymbol{h}) = 0$ and $\gamma_{22}(\boldsymbol{h}) =0$ due to inequalities (4) and (5). Suppose now that $\gamma_{11}(\boldsymbol{h}), \gamma_{22}(\boldsymbol{h}) \neq 0$ . Letting $k \rightarrow \infty$ in inequalities (4) and (5) yields $\gamma_{11}(\boldsymbol{h}) = \gamma_{22}(\boldsymbol{h})$ . Inserting this into inequality (5) gives

\[\gamma_{22}(\boldsymbol{h}) \le \dfrac{1}{1-{{1}/{k}}}\gamma_{12}(\boldsymbol{h}) \le \dfrac{1}{1-{{1}/{k}}}\gamma_{22}(\boldsymbol{h})\]

and consequently the result for $k \rightarrow \infty$ .

Corollary 1 can also be proved by means of a result in [Reference Oesting, Schlather and Friederichs21]. In fact, Theorem 1 enables us to reproduce their result, which was originally derived in a stochastic manner, by a direct proof.

Corollary 2. Let ${\boldsymbol \gamma}\,:\, {{\mathbb{R}}}^d \rightarrow {{\mathbb{R}}}^{m \times m}$ be a pseudo cross-variogram. Then ${\boldsymbol \gamma}$ fulfils

\[\Bigl(\sqrt{\gamma_{ii}(\boldsymbol{h})}-\sqrt{\gamma_{ij}(\boldsymbol{h})}\Bigr)^2 \le \gamma_{ij}(\textbf{0}), \quad \boldsymbol{h} \in {{\mathbb{R}}}^d, \quad i,j=1,\ldots,m.\]

Proof. Without loss of generality, assume $m=2$ . We present a proof for $i=1, j=2$ and $\gamma_{11}(\boldsymbol{h})$ , $\gamma_{12}(\boldsymbol{h}) > 0$ . Then, for $n=2$ , $\boldsymbol{x}_{\textbf{1}}=\boldsymbol{h}$ , $\boldsymbol{x}_{\textbf{2}}=\textbf{0}$ , $\boldsymbol{a}_{\textbf{1}},\boldsymbol{a}_{\textbf{2}} \in {{\mathbb{R}}}^2$ with $\textbf{1}_{\textbf{2}}^\top \sum_{k=1}^2 \boldsymbol{a}_{\boldsymbol{k}}=0$ , we have

(6) \begin{align} 0 &\ge a_{11}a_{21} \gamma_{11}(\boldsymbol{h})+a_{12}a_{22}\gamma_{22}(\boldsymbol{h}) \notag \\ &\quad + a_{11}a_{22}\gamma_{12}(\boldsymbol{h}) + a_{12}a_{21}\gamma_{21}(\boldsymbol{h}) + (a_{11}a_{12}+a_{21}a_{22})\gamma_{12}(\textbf{0}).\end{align}

Assuming $a_{12}=0$ , $a_{22}>0$ and $a_{11} + a_{22} = -a_{21} >0$ , inequality (6) simplifies to

(7) \begin{align} \gamma_{12}(\textbf{0}) &\ge -\dfrac{a_{11}}{a_{22}} \gamma_{11}(\boldsymbol{h}) + \dfrac{a_{11}}{a_{11}+a_{22}}\gamma_{12}(\boldsymbol{h}) \notag \\&= -x \gamma_{11}(\boldsymbol{h}) + \dfrac x{1+x}\gamma_{12}(\boldsymbol{h})\end{align}

for $x \,:\!=\, {{a_{11}}/{a_{22}}}$ . Maximization of the function

\[x \mapsto -x \gamma_{11}(\boldsymbol{h}) + \dfrac x{1+x}\gamma_{12}(\boldsymbol{h}),\quad x > -1,\]

leads to

\[x^{\ast} = \sqrt{\dfrac{\gamma_{12}(\boldsymbol{h})}{\gamma_{11}(\boldsymbol{h})}} -1.\]

Inserting $x^\ast$ into (7) gives

\begin{align*} \gamma_{12}(\textbf{0}) &\ge -\biggl(\sqrt{\dfrac{\gamma_{12}(\boldsymbol{h})}{\gamma_{11}(\boldsymbol{h})}}-1\biggr) \gamma_{11}(\boldsymbol{h}) + \left( \dfrac {\sqrt{\frac{\gamma_{12}(\boldsymbol{h})}{\gamma_{11}(\boldsymbol{h})}} -1}{1+\sqrt{\frac{\gamma_{12}(\boldsymbol{h})}{\gamma_{11}(\boldsymbol{h})}} -1} \right)\gamma_{12}(\boldsymbol{h}) \\ &= \bigl(\sqrt{\gamma_{11}(\boldsymbol{h})}-\sqrt{\gamma_{12}(\boldsymbol{h})}\bigr)^2.\end{align*}

3. A Schoenberg-type characterization

The stochastically motivated proof of Theorem 1 contains an important relation between matrix-valued positive definite kernels and conditionally negative definite functions we have not yet emphasized. Due to its significance, we formulate it in a separate lemma. As readily seen, the assumption on the main diagonal stemming from our consideration of pseudo cross-variograms can be dropped, resulting in the matrix-valued version of Lemma 3.2.1 in [Reference Berg, Christensen and Ressel2].

Lemma 1. Let ${\boldsymbol \gamma}\,:\, {{\mathbb{R}}}^d \rightarrow {{\mathbb{R}}}^{m \times m}$ be a matrix-valued function with $\gamma_{ij}(\boldsymbol{h})=\gamma_{ji}({-}\boldsymbol{h})$ , $i,j=1,\ldots,m$ . Define

\[ \boldsymbol{C}_{\boldsymbol{k}}(\boldsymbol{x},\boldsymbol{y})\,:\!=\, \boldsymbol{\gamma_{\boldsymbol{k}}}(\boldsymbol{x})\textbf{1}_{\boldsymbol{m}}^\top + \textbf{1}_{\boldsymbol{m}}\boldsymbol{\gamma}_{\boldsymbol{k}}^\top(\boldsymbol{y}) -{\boldsymbol \gamma}(\boldsymbol{x} -\boldsymbol{y})-\gamma_{kk}(\textbf{0})\textbf{1}_{\boldsymbol{m}}\mathbf{1}_{\boldsymbol{m}}^\top \]

with $\boldsymbol{\gamma_{\boldsymbol{k}}}(\boldsymbol{h})=(\gamma_{1k}(\boldsymbol{h}),\ldots,\gamma_{mk}(\boldsymbol{h}))^\top$ , $k \in \{1,\ldots,m\}$ . Then $\boldsymbol{C}_{\boldsymbol{k}}$ is a positive definite matrix-valued kernel for $k \in \{1,\ldots,m\}$ if and only if ${\boldsymbol \gamma}$ is conditionally negative definite. If $\gamma_{kk}(\textbf{0}) \ge 0$ for $k=1,\ldots,m$ , then

\[ \tilde{\boldsymbol{C}_{\boldsymbol{k}}}(\boldsymbol{x},\boldsymbol{y})\,:\!=\, \boldsymbol{\gamma_{\boldsymbol{k}}}(\boldsymbol{x})\textbf{1}_{\boldsymbol{m}}^\top + \textbf{1}_{\boldsymbol{m}}\boldsymbol{\gamma}_{\boldsymbol{k}}^\top(\boldsymbol{y}) -{\boldsymbol \gamma}(\boldsymbol{x} -\boldsymbol{y}) \]

is a positive definite matrix-valued kernel for $k \in \{1,\ldots,m\}$ if and only if ${\boldsymbol \gamma}$ is conditionally negative definite.

The kernel construction in Lemma 1 leads to a matrix-valued version of Schoenberg’s theorem [Reference Berg, Christensen and Ressel2].

Proof of Theorem 2.Assume that ${\boldsymbol \gamma}$ is conditionally negative definite. Then

\[(\boldsymbol{x},\boldsymbol{y}) \mapsto \boldsymbol{\gamma}_{\textbf{1}}(\boldsymbol{x})\textbf{1}_{\boldsymbol{m}}^\top + \textbf{1}_{\boldsymbol{m}}\boldsymbol{\gamma}_{\textbf{1}}^\top(\boldsymbol{y}) -{\boldsymbol \gamma}(\boldsymbol{x} -\boldsymbol{y}) -\gamma_{11}(\textbf{0})\textbf{1}_{\boldsymbol{m}}\textbf{1}_{\boldsymbol{m}}^\top\]

is a positive definite kernel due to Lemma 1. Since positive definite matrix-valued kernels are closed with regard to sums, Hadamard products, and pointwise limits (see e.g. [Reference Schlather27]), the kernel

\begin{align*}(\boldsymbol{x},\boldsymbol{y}) &\mapsto \exp^\ast\bigl(t\boldsymbol{\gamma}_{\textbf{1}}(\boldsymbol{x})\textbf{1}_{\boldsymbol{m}}^\top + t\textbf{1}_{\boldsymbol{m}}\boldsymbol{\gamma}_{\textbf{1}}^\top(\boldsymbol{y}) -t{\boldsymbol \gamma}(\boldsymbol{x} -\boldsymbol{y})-t\gamma_{11}(\textbf{0})\textbf{1}_{\boldsymbol{m}}\textbf{1}_{\boldsymbol{m}}^\top\bigr) \\&=\exp({-}t\gamma_{11}(\textbf{0}))\exp^\ast\bigl(t\boldsymbol{\gamma}_{\textbf{1}}(\boldsymbol{x})\textbf{1}_{\boldsymbol{m}}^\top + t\textbf{1}_{\boldsymbol{m}}\boldsymbol{\gamma}_{\textbf{1}}^\top(\boldsymbol{y}) -t{\boldsymbol \gamma}(\boldsymbol{x} -\boldsymbol{y})\bigr)\end{align*}

is again positive definite for all $t >0$ . The same holds true for the kernel

\[(\boldsymbol{x},\boldsymbol{y}) \mapsto \exp^*\bigl({-}t\boldsymbol{\gamma}_{\textbf{1}}(\boldsymbol{x})\textbf{1}_{\boldsymbol{m}}^\top - t\textbf{1}_{\boldsymbol{m}}\boldsymbol{\gamma}_{\textbf{1}}^\top(\boldsymbol{y})\bigr),\]

since

\[\exp\bigl({-}t\boldsymbol{\gamma}_{\textbf{1}}(\boldsymbol{x})\textbf{1}_{\boldsymbol{m}}^\top - t\textbf{1}_{\boldsymbol{m}}\boldsymbol{\gamma}_{\textbf{1}}^\top(\boldsymbol{y})\bigr)_{ij}=\exp({-}t\gamma_{i1}(\boldsymbol{x}))\exp({-}t\gamma_{j1}(\boldsymbol{y})), \quad i,j=1,\ldots,m,\]

with the product separable structure implying positive definiteness. Again using the stability of positive definite kernels under Hadamard products, the first part of the assertion follows.

Assume now that $\exp^\ast({-}t{\boldsymbol \gamma})$ is a positive definite function for all $t>0$ . Then

\[\exp({-}t\gamma_{ij}(\boldsymbol{h}))=\exp({-}t\gamma_{ji}({-}\boldsymbol{h})),\]

and thus

\[\biggl( \dfrac{1 - {{e}}^{-t\gamma_{ij}}}t\biggr)_{i,j=1,\ldots,m}= \dfrac{\textbf{1}_{\boldsymbol{m}} \textbf{1}_{\boldsymbol{m}}^\top - \exp^\ast({-}t{\boldsymbol \gamma})}t\]

is a conditionally negative definite function. The assertion follows for $t\rightarrow 0$ .

Combining Theorems 1 and 2, and recalling that the classes of matrix-valued positive definite functions and covariance functions for multivariate random fields coincide, we immediately get the following characterization of pseudo cross-variograms.

Corollary 3 establishes a direct link between matrix-valued correlation functions and pseudo cross-variograms. Together with Corollary 1, it shows that the cross-variograms for which Theorem 10 in [Reference Ma14] holds are necessarily of the form (3), and it explains the findings in the first part of Remark 2 in [Reference Schlather27].

As demonstrated in Theorem 3.2.3 in [Reference Berg, Christensen and Ressel2], Schoenberg’s theorem can be further generalized for conditionally negative definite functions with non-negative components in terms of componentwise Laplace transforms. Here we present the corresponding matrix-valued version explicitly for clarity, and combine it with our previous results concerning pseudo cross-variograms. With regard to Remark 3, and since we have already presented the matrix-valued proof of Theorem 2, we omit the proof here.

Theorem 3. Let $\mu$ be a probability measure on $[0,\infty)$ such that

\[ 0 < \int_0^\infty s \,{{d}} \mu(s) < \infty. \]

Let $\mathcal{L}$ denote its Laplace transform, that is,

\[ \mathcal{L}\mu(x)=\int_0^\infty \exp({-}sx) \,{{d}} \mu(s),\quad x \in [0,\infty). \]

Then ${\boldsymbol \gamma}\,:\, {{\mathbb{R}}}^d \rightarrow [0,\infty)^{m \times m}$ is conditionally negative definite if and only if $(\mathcal{L}\mu(t\gamma_{ij}))_{i,j=1,\ldots,m}$ is positive definite for all $t >0$ . In particular, ${\boldsymbol \gamma}$ is a pseudo cross-variogram if and only if $(\mathcal{L}\mu(t\gamma_{ij}))_{i,j=1,\ldots,m}$ is an m-variate correlation function for all $t >0$ .

Corollary 4. Let ${\boldsymbol \gamma}\,:\, {{\mathbb{R}}}^d \rightarrow [0,\infty)^{m \times m}$ be a matrix-valued function. If ${\boldsymbol \gamma}$ is a pseudo cross-variogram, then for all $\lambda >0$ , the function $\boldsymbol{C}\,:\, {{\mathbb{R}}}^d \rightarrow {{\mathbb{R}}}^{m \times m}$ with

(8) \begin{equation} \boldsymbol{C}(\boldsymbol{h})=\bigl((1 + t\gamma_{ij}(\boldsymbol{h}))^{-\lambda}\bigr)_{i,j=1,\ldots,m}, \quad \boldsymbol{h} \in {{\mathbb{R}}}^d, \end{equation}

is a correlation function of an m-variate random field for all $t >0$ . Conversely, if a $\lambda >0$ exists such that $C_{ii}(\textbf{0})=1$ , $i=1,\ldots,m$ , and such that $\boldsymbol{C}$ is positive definite for all $t >0$ , then ${\boldsymbol \gamma}$ is a pseudo cross-variogram.

Proof. Choose

\[ \mu({{d}} s)=\dfrac1{\Gamma(\lambda)}\exp({-}s)s^{\lambda-1} \unicode{x1d7d9}_{(0,\infty)}(s) \,{{d}} s \]

in Theorem 3.

Similarly to Theorem 3, we can translate the ‘univariate’ result that Bernstein functions operate on real-valued conditionally negative definite functions [Reference Berg, Christensen and Ressel2] to the matrix-valued case, which can thus be used to derive novel pseudo cross-variograms from known ones. Again, we omit the proof for the reasons given above.

4. Multivariate versions of Gneiting’s space–time model

Schoenberg’s result is often an integral part of proving the validity of univariate covariance models. Here we use its matrix-valued counterparts derived in the previous section to naturally extend covariance models of Gneiting type to the multivariate case.

Gneiting’s original space–time model is a univariate covariance function on ${{\mathbb{R}}}^d\times {{\mathbb{R}}}$ defined via

(9) \begin{equation} G(\boldsymbol{h},u)= \dfrac1{\psi(\lvert u\rvert^2)^{d/2}}\varphi\biggl(\dfrac{\lVert \boldsymbol{h}\rVert^2}{\psi(\lvert u\rvert^2)}\biggr), \quad (\boldsymbol{h},u) \in {{\mathbb{R}}}^d \times {{\mathbb{R}}},\end{equation}

where $\psi\,:\, 0, \infty) \rightarrow (0,\infty)$ is the continuous extension of a Bernstein function, and $\varphi\,:\, [0,\infty) \rightarrow [0,\infty)$ is the continuous extension of a bounded completely monotone function [Reference Gneiting10]. For convenience, we simply speak of bounded completely monotone functions henceforth. Model (9) is very popular in practice due to its versatility and ability to model space–time interactions; see [Reference Porcu, Furrer and Nychka24] for a list of several applications. Its special structure has attracted and still attracts interest from a theoretical perspective as well, resulting in several extensions and refinements of the original model (9); see e.g. [Reference Menegatto17], [Reference Menegatto, Oliveira and Porcu18], [Reference Porcu, Bevilacqua and Genton23], [Reference Qadir and Sun25], [Reference Zastavnyi and Porcu32]. Only recently, specific simulation methods have been proposed [Reference Allard, Emery, Lacaux and Lantuéjoul1] for the so-called extended Gneiting class, a special case of [Reference Zastavnyi and Porcu32, Theorem 2.1],

\begin{equation*} G(\boldsymbol{h},\boldsymbol{u})= \dfrac1{(1+\gamma(\boldsymbol{u}))^{d/2}}\varphi\biggl(\dfrac{\lVert \boldsymbol{h}\rVert^2}{1+\gamma(\boldsymbol{u})}\biggr), \quad (\boldsymbol{h},\boldsymbol{u}) \in {{\mathbb{R}}}^d \times {{\mathbb{R}}}^l,\end{equation*}

with $\gamma$ denoting a continuous variogram. One of these methods is based on an explicit construction of a random field, where the continuity assumption on $\gamma$ is not needed [Reference Allard, Emery, Lacaux and Lantuéjoul1], and which can be directly transferred to the multivariate case via pseudo cross-variograms.

Theorem 4. Let R be a non-negative random variable with distribution $\mu$ , $\boldsymbol{\Omega} \sim N(\textbf{0},\mathbf{1}_{d \times d})$ with $\mathbf{1}_{d \times d} \in {{\mathbb{R}}}^{d \times d}$ denoting the identity matrix, $U \sim U(0,1)$ , $\Phi\sim U(0,2\pi)$ , and let $\boldsymbol{W}$ be a centred, m-variate Gaussian random field on ${{\mathbb{R}}}^l$ with pseudo cross-variogram ${\boldsymbol \gamma}$ , all independent. Then the m-variate random field $\boldsymbol{Z}$ on ${{\mathbb{R}}}^d \times {{\mathbb{R}}}^l$ defined via

\[ Z_i(\boldsymbol{x},\boldsymbol{t})=\sqrt{-2\log(U)}\cos\biggl(\sqrt{2R}\langle\boldsymbol{\Omega},\boldsymbol{x}\rangle + \dfrac{\lVert \boldsymbol{\Omega} \rVert}{\sqrt{2}}W_i(\boldsymbol{t})+\Phi \biggr),\!\!\quad (\boldsymbol{x},\boldsymbol{t}) \in {{\mathbb{R}}}^d\times {{\mathbb{R}}}^l,\quad i =1,\ldots,m, \]

has the extended Gneiting-type covariance function

(10) \begin{equation} G_{ij}(\boldsymbol{h},\boldsymbol{u})= \dfrac1{(1+\gamma_{ij}(\boldsymbol{u}))^{d/2}}\varphi\biggl(\dfrac{\lVert \boldsymbol{h}\rVert^2}{1+\gamma_{ij}(\boldsymbol{u})}\biggr),\quad (\boldsymbol{h},\boldsymbol{u}) \in {{\mathbb{R}}}^d\times {{\mathbb{R}}}^l, \quad i,j=1,\ldots,m,\end{equation}

where $\varphi$ denotes a bounded completely monotone function.

Proof. The proof follows the lines of the proof of Theorem 3 in [Reference Allard, Emery, Lacaux and Lantuéjoul1]. In the multivariate case, the cross-covariance function reads

\begin{align*}{{{ Cov}}}(Z_i(\boldsymbol{x},\boldsymbol{t}),Z_j(\boldsymbol{y},\boldsymbol{s}))&= {{\mathbb{E}}} \cos\biggl(\sqrt{2R}\langle\boldsymbol{\Omega},\boldsymbol{x}-\boldsymbol{y}\rangle + \dfrac{\lVert \boldsymbol{\Omega}\rVert}{\sqrt{2}}(W_i(\boldsymbol{t})-W_j(\boldsymbol{s}))\biggr),\end{align*}

$(\boldsymbol{x},\boldsymbol{t}), (\boldsymbol{y},\boldsymbol{s}) \in {{\mathbb{R}}}^d \times {{\mathbb{R}}}^l, i,j=1,\ldots,m$ . Due to the assumptions, $W_i(\boldsymbol{t})-W_j(\boldsymbol{s})$ is a Gaussian random variable with mean zero and variance $2\gamma_{ij}(\boldsymbol{t}-\boldsymbol{s})$ . Proceeding further as in [Reference Allard, Emery, Lacaux and Lantuéjoul1] gives the result.

Theorem 4 provides a multivariate extension of the extended Gneiting class, and lays the foundations for a simulation algorithm for an approximately Gaussian random field with the respective cross-covariance function; see [Reference Allard, Emery, Lacaux and Lantuéjoul1]. The existence of a Gaussian random field with a preset pseudo cross-variogram ${\boldsymbol \gamma}$ and the possibility of sampling from it are ensured by Theorem 1 and Lemma 1, respectively.

Due to our results in previous sections, Theorem 4 can easily be generalized further, replacing $d/2$ in the denominator in equation (10) with a general parameter $r \ge d/2$ .

Corollary 5. Let ${\boldsymbol \gamma}\,:\, {{\mathbb{R}}}^l \rightarrow {{\mathbb{R}}}^{m \times m}$ be a pseudo cross-variogram. Then the function $\boldsymbol{G}\,:\, {{\mathbb{R}}}^d \times {{\mathbb{R}}}^l \rightarrow {{\mathbb{R}}}^{m \times m}$ with

(11) \begin{equation} G_{ij}(\boldsymbol{h},\boldsymbol{u})= \dfrac1{(1+\gamma_{ij}(\boldsymbol{u}))^{r}}\varphi\biggl(\dfrac{\lVert \boldsymbol{h}\rVert^2}{1+\gamma_{ij}(\boldsymbol{u})}\biggr), \quad (\boldsymbol{h},\boldsymbol{u}) \in {{\mathbb{R}}}^d\times {{\mathbb{R}}}^l, \quad i,j=1,\ldots,m,\end{equation}

is positive definite for $r \ge {{d}/{2}}$ and a bounded completely monotone function $\varphi$ .

Even further refinements of Corollary 5 are possible. We can replace $\textbf{1}_{\boldsymbol{m}}\textbf{1}_{\boldsymbol{m}}^\top + {\boldsymbol \gamma}$ in (11) with general conditionally negative definite matrix-valued functions, but for a subclass of completely monotone functions, the so-called generalized Stieltjes functions of order $\lambda$ . This leads to a multivariate version of a result in [Reference Menegatto17]. A bounded generalized Stieltjes function $S\,:\, (0,\infty) \rightarrow [0,\infty)$ of order $\lambda >0$ has a representation

\[ S(x) = a + \int_0^\infty \dfrac1{(x + v)^\lambda}\,{{d}} \mu(v), \quad x>0, \]

where $a\ge 0$ and the so-called Stieltjes measure $\mu$ is a positive measure on $(0,\infty)$ , such that $\int_{(0,\infty)}v^{-\lambda}\,{{d}} \mu(v) < \infty$ [Reference Menegatto17]. As for completely monotone functions, in the following we do not distinguish between a generalized Stieltjes function and its continuous extension. Several examples of generalized Stieltjes functions can be found in [Reference Berg, Koumandos and Pedersen3] and [Reference Menegatto17].

Theorem 5. Let $S_{ij}, i,j=1,\ldots,m$ , be generalized Stieltjes functions of order $\lambda >0$ . Let the associated Stieltjes measures have densities $\varphi_{ij}$ such that $(\varphi_{ij}(v))_{i,j=1,\ldots,m}$ is a symmetric positive semidefinite matrix for all $v > 0$ . Let

\[ \boldsymbol{g}\,:\, {{\mathbb{R}}}^d \rightarrow [0,\infty)^{m \times m},\quad \boldsymbol{f}\,:\, {{\mathbb{R}}}^l \rightarrow (0,\infty)^{m \times m} \]

be conditionally negative definite functions. Then the function $\boldsymbol{G}\,:\, {{\mathbb{R}}}^d\times {{\mathbb{R}}}^l \rightarrow {{\mathbb{R}}}^{m \times m}$ with

\[ G_{ij}(\boldsymbol{h},\boldsymbol{u})= \dfrac1{f_{ij}(\boldsymbol{u})^r}S_{ij}\biggl(\dfrac{g_{ij}(\boldsymbol{h})}{f_{ij}(\boldsymbol{u})}\biggr), \quad (\boldsymbol{h},\boldsymbol{u}) \in {{\mathbb{R}}}^d\times{{\mathbb{R}}}^l, \quad i,j=1,\ldots,m, \]

is an m-variate covariance function for $r \ge \lambda$ .

Proof. We follow the proof in [Reference Menegatto17]. It holds that

\begin{align*} G_{ij}(\boldsymbol{h},\boldsymbol{u})&= \dfrac a{f_{ij}(\boldsymbol{u})^r} + \dfrac 1{f_{ij}(\boldsymbol{u})^{r-\lambda}} \int_0^\infty \dfrac1{(g_{ij}(\boldsymbol{h}) + vf_{ij}(\boldsymbol{u}))^\lambda}\varphi_{ij}(v) \,{{d}} v .\end{align*}

The function $ x \mapsto {{1}/{x^\alpha}}$ is completely monotone for $\alpha \ge 0$ and thus the Laplace transform of a measure on $[0,\infty)$ [Reference Schilling, Song and Vondracek26, Theorem 1.4]. Therefore $(1/f_{ij}^r)_{i,j=1,\ldots,m}$ and $(1/f_{ij}^{r-\lambda})_{i,j=1,\ldots,m}$ are positive definite functions due to Theorem 2 as mixtures of positive definite functions. Furthermore, we have

\[\dfrac1{(g_{ij}(\boldsymbol{h}) + vf_{ij}(\boldsymbol{u}))^\lambda}= \dfrac1{\Gamma(\lambda)}\int_0^\infty {{e}}^{-sg_{ij}(\boldsymbol{h})}{{e}}^{-svf_{ij}(\boldsymbol{u})}s^{\lambda -1} \,{{d}} s.\]

The functions $\bigl({{e}}^{-sg_{ij}(\boldsymbol{h})}\bigr)_{i,j=1,\ldots,m}$ and $\bigl({{e}}^{-svf_{ij}(\boldsymbol{u})}\bigr)_{i,j=1,\ldots,m}$ are again positive definite due to Theorem 2 for all $s,v >0$ , and so is their componentwise product. Since positive definite functions are closed under integration,

\[\biggl(\dfrac1{(g_{ij}(\boldsymbol{h}) + vf_{ij}(\boldsymbol{u}))^\lambda}\biggr)_{i,j=1,\ldots,m}\]

is positive definite for all $v > 0$ . Therefore the function

\[\biggl(\dfrac1{(g_{ij}(\boldsymbol{h}) + vf_{ij}(\boldsymbol{u}))^\lambda}\varphi_{ij}(v)\biggr)_{i,j=1,\ldots,m}\]

is also positive definite for all $v > 0$ . Combining and applying the above arguments shows our claim.

Theorem 5 provides a very flexible model. In a space–time framework, it allows for different cross-covariance structures in both space and time, and it does not require assumptions such as continuity of the conditionally negative definite functions involved, or isotropy, which distinguishes it from the multivariate Gneiting-type models presented in [Reference Bourotte, Allard and Porcu4] and [Reference Guella12], respectively.