Is the spherical covariance function not positive definite for d > 3?

I read in a textbook (Japanese one) that the spherical covariance function is only valid for dimensions $$d = 1,$$ $$2,$$ and $$3.$$ I have the following questions:

1. Does that mean the spherical covariance function is not positive definite for $$d\gt 3?$$
2. Is there any material which contains a proof of (1)?

There are at least two conventional meanings of "spherical covariance function." Based on the conclusion you want to draw, we can infer you are referring to the family of functions $$G:\mathbb{R}^n\to \mathbb{R}$$ based on

$$G(\mathbf{h}) = 1 - \frac{3|\mathbf{h}|}{2} + \frac{|\mathbf{h}|^3}{2},\ 0 \le |\mathbf{h}| \le 1,$$

and otherwise $$G=0.$$ ("Based on" means we can ignore scale factors in both $$\mathbf h$$ and $$G$$ without any loss of generality merely by choosing our units of measurement appropriately.)

A positive-(semi)definite covariance $$C$$ is one that satisfies

$$\sum_{j,k} a_j C(\mathbf{y}_j - \mathbf{y}_k) \bar{a_k} \ge 0$$

for any collection of locations $$\mathbf{y}_i \in \mathbb{R}^n, i=1,2,\ldots, m$$ and all possible complex numbers $$(a_i), i=1,2,\ldots, m.$$ In other words, the covariance matrix associated with any set of locations is positive-semidefinite.

Bochner's Theorem asserts that all positive-definite covariances arise as the Fourier transforms of finite positive measures $$F$$ given by

$$\hat F(\mathbf{h}) = \int_{\mathbb{R}^n} e^{i\mathbf{h}\cdot \mathbf{x}} F(\mathrm{d}\mathbf{x}).$$

Indeed, by interchanging summation and integration we may check that

\eqalign{ \sum_{j,k} a_j C(\mathbf{y}_j - \mathbf{y}_k) \bar{a_k} &= \sum_{j,k}\int_{\mathbf{R}^n}a_j e^{i(\mathbf{y}_j - \mathbf{y}_k)\cdot \mathbf{x}} \bar{a}_kF(\mathrm{d}\mathbf{x}) \\ &= \int_{\mathbf{R}^n} \sum_j a_j e^{i\mathbf{y}_j\cdot \mathbf{x}}\ \sum_k \overline{e^{i\mathbf{y}_k \cdot\mathbf{x}}a_k}\ F(\mathrm{d}\mathbf{x}) \\ &= \int_{\mathbf{R}^n} \left|\sum_j a_j e^{i\mathbf{y}_j\cdot \mathbf{x}}\ \right|^2\ F(\mathrm{d}\mathbf{x}) \ge 0. }

Because (for spherical covariances) the expression $$C(\mathbf{y}_j - \mathbf{y}_k)$$ depends only on the separation distances $$r_{jk}=|\mathbf h_{jk}| = |\mathbf{y}_j - \mathbf{y}_k|,$$ the Fourier transform can be inverted (to find $$F$$ from $$G$$) with a single integration. By doing the calculation we find the following measures $$F(\mathrm{d}\mathbf{x}) = F(r)\mathrm{d}r \mathrm{d}\Omega$$ in spherical coordinates $$(r, \Omega)$$ for $$n=3,4:$$

$$n=3:\ F(r) \propto r^{-6}\left(r \cos(r/2) - 2 \sin(r/2)\right)^2 \ge 0.$$ $$n=4:\ F(r) \propto r^{-6}\left(\left(r^2+5\right) (\pmb{H}_0(r) J_1(r)-\pmb{H}_1(r) J_0(r))+30 r (r J_0(r)-3 J_1(r))\right),$$

an algebraic combination of Bessel J functions $$J_i$$ and Struve H functions $$\pmb{H}_i.$$ Because some of the possible values of this density are negative, the spherical covariance function for $$n=4$$ is not positive-definite.

We are actually done: we don't have to consider all possible dimensions. The reason is that some configurations of points in $$\mathbb{R}^n$$ actually lie within lower-dimensional subspaces. Consequently, it is immediate from the definition of positive-definite functions that

1. When a covariance in $$\mathbb{R}^n$$ is positive-definite, it remains positive-definite in all lower dimensions.

2. When a covariance in $$\mathbb{R}^n$$ fails to be positive-definite, it fails in all higher dimensions.

Reference

This post broadly follows the exposition in a set of Duke University lecture notes (unattributed and undated) found at https://www2.stat.duke.edu/courses/Fall01/sta293/lecs/covar.pdf. Its bibliography covers the classics: Yaglom, Matern, Ripley, Cressie.

• Which book should I read to understand the derivation of (1) or (4) from the first line in the lecture notes? §1.1 is too brief for me.
– Jiro
Jul 22 '19 at 7:08
• The reference to "Watson 1944" is almost surely the second edition of Watson's Treatise on the Theory of Bessel Functions, available online at ia802601.us.archive.org/30/items/…. For an easier introduction see Whittaker and Watson, A Course of Modern Analysis (1927 edition, Cambridge U. Press).
– whuber
Jul 22 '19 at 19:59
• Thank you for letting me know about those books. But I can't follow changes of variables before the line involving a Bessel function. In §1.1, I see that $d^nx = \frac{2\pi^{n/2}}{\Gamma(n/2)}r^{n-1}dr\,d\sigma$ and $∑_{j=1}^n σ_j^2 = 1$, but the last sentence of that section doesn't make sense to me at all. (I know very basic definitions and properties of Bernoulli and Dirichlet distributions.)
– Jiro
Jul 23 '19 at 5:50
• I take "$\sigma_j$" to be coordinate $j$ on the unit sphere $S^{n-1}\subset\mathbb{R}^n.$ For the uniform distribution on $S^n,$ $(\sigma_j+1)/2$ has a Beta$((n-1)/2,(n-1)/2)$ distribution: stats.stackexchange.com/a/85977/919. The claim in the paper is correct only if $\sigma_j$ is taken to be the absolute value of the coordinate. In that case the $\sigma_j^2$ have the same distribution as $x_j^2/(x_1^2+\cdots+x_n^2)$ for iid standard normal variables $x_i,$ which is easily seen to be a Gamma$(1/2)$ distribution, whence $(\sigma_j)^2$ has a Dirichlet$(1/2,\ldots,1/2)$ distribution.
– whuber
Jul 23 '19 at 14:51
• Thank you very much! Now I understand that $u \in [0, 1]$ is defined as $u = \sigma_h^2$ and its distribution is $\mathrm{Beta} (\frac{1}{2}, \frac{n-1}{2})$.
– Jiro
Jul 25 '19 at 8:20