Isotropic Covariance Matrix Functions On All Spheres

Abstract

This paper reviews and introduces characterizations of the covariance function on all spheres that is isotropic and continuous, and characterizations of the covariance matrix function on all spheres whose entries are isotropic and continuous. These characterizations are used to derive some covariance (matrix) structures on all spheres, with certain polynomials obtained, besides some rational, (negative) power, and logarithmic models.

Appendices

Appendix A: Proof of Theorem 1

Statement (i) in Theorem 1 implying (ii) was shown by Schoenberg (1942), and Bingham (1973) proved that condition (ii) was sufficient for statement (i); see also Christensen and Ressel (1978).

Under the assumption (ii), (iii) is established after defining

$$\begin{aligned} g(x) = \sum _{n=0}^\infty b_n x^n, \quad -1 \le x \le 1, \end{aligned}$$

which is continuous on $[-1, 1]$ and absolutely monotone on $[0, 1]$. Clearly, both $g(x)+g(-x)$ and $g(x)-g(-x)$ are absolutely monotone on $[0, 1]$.

Conversely, let $g(x)$ satisfy the conditions in (iii). Since both $g(x)+g(-x)$ and $g(x)-g(-x)$ are absolutely monotone on $[0, 1]$, they possess the Taylor series

$$\begin{aligned} g(x)+g(-x) = \sum _{n=0}^\infty \frac{b_{1n}}{n!} x^n, \quad 0 \le x \le 1, \end{aligned}$$

and

$$\begin{aligned} g(x)-g(-x) = \sum _{n=0}^\infty \frac{b_{2n}}{n!} x^n, \quad 0 \le x \le 1, \end{aligned}$$

respectively, where

$$\begin{aligned} b_{1n}&= \left. \frac{d^n}{dx^n} (g(x)+g(-x)) \right| _{x=0} = \left\{ \begin{array}{ll} 2 g^{(n)}(0), &{} \quad n {\hbox { is zero or even}}, \\ 0, &{} \quad n {\hbox { is odd}}, \end{array} \right. \\ b_{2n}&= \left. \frac{d^n}{dx^n} (g(x)-g(-x)) \right| _{x=0} = \left\{ \begin{array}{ll} 0, &{} \quad n {\hbox { is zero or even}}, \\ 2 g^{(n)}(0), &{} \quad n {\hbox { is odd}}, \end{array} \right. \end{aligned}$$

according to Theorem 3a of Widder (1946), page 146. Thus,

$$\begin{aligned} g(x)&= \frac{ g(x)+g(-x)}{2} + \frac{ g(x)-g(-x)}{2} = \sum _{n=0}^\infty \frac{b_{1n}+b_{2n}}{2 n!} x^n\\&= \sum _{n=0}^\infty b_n x^n, \quad 0 \le x \le 1,\\ g(-x)&= \frac{ g(x)+g(-x)}{2} - \frac{ g(x)-g(-x)}{2}\\&= \sum _{n=0}^\infty \frac{b_{1n}-b_{2n}}{2 n!} x^n = \sum _{n=0}^\infty b_n (-x)^n, \quad 0 \le x \le 1, \end{aligned}$$

where

$$\begin{aligned} b_n = \frac{g^{(n)}(0)}{n!}, \quad n = 0, 1, \ldots , \end{aligned}$$

is a summable sequences of nonnegative numbers. In other words, $g(x)$ possesses the Taylor series

$$\begin{aligned} g(x) = \sum _{n=0}^\infty \frac{g^{(n)}(0)}{n!} x^n, \quad -1 \le x \le 1, \end{aligned}$$

so that $C(\vartheta ) = g( \cos \vartheta )$ is of the form (3) in Statement (ii).

To establish the equivalency between (iii) and (iv), we apply identity (14) to rewrite $C(\vartheta )$ as

$$\begin{aligned} C(\vartheta ) = C \left( \frac{\pi }{2} -\arcsin (\cos \vartheta ) \right) , \quad \vartheta \in [0, \pi ], \end{aligned}$$

so that $g(x)$ is defined by

$$\begin{aligned} g(x) = C \left( \frac{\pi }{2} -\arcsin x \right) , \quad x \in [-1, 1]. \end{aligned}$$

Appendix B: Proof of Theorem 2

• (i) Recall that the composition of two absolutely monotone functions is also absolutely monotone (Theorem 2a of Widder (1946), page 145). As the composition of two absolutely monotone functions $C\left( \frac{\pi }{2}-x \right) +C\left( \frac{\pi }{2}+x \right)$ and $\arcsin x$, $C\left( \frac{\pi }{2}-\arcsin x \right) +C\left( \frac{\pi }{2}+\arcsin x \right)$ is absolutely monotone on $[0, 1]$. Similarly, $C\left( \frac{\pi }{2}-\arcsin x \right) -C\left( \frac{\pi }{2}+\arcsin x \right)$ is absolutely monotone on $[0, 1]$.

• (ii) It suffices to verify that the conditions of Theorem 2 (i) are satisfied. Since $C(\vartheta )$ is completely monotone on $[0, \pi ]$, for each natural number $n$, $(-1)^{n} C^{(n)}(\vartheta )$ is a nonnegative and decreasing function on $[0, \pi ]$, and thus

  $$\begin{aligned} (-1)^n C^{(n)} \left( \frac{\pi }{2} -\vartheta \right) -(-1)^n C^{(n)} \left( \frac{\pi }{2}+\vartheta \right) \ge 0, \quad 0 \le \vartheta \le \frac{\pi }{2}. \end{aligned}$$

  For an even $n$, we obtain

  $$\begin{aligned}&\frac{d^n}{d x^n} \left( C \left( \frac{\pi }{2} -x \right) - C \left( \frac{\pi }{2} +x \right) \right) \\&\quad = (-1)^n C^{(n)} \left( \frac{\pi }{2} -x \right) - C^{(n)} \left( \frac{\pi }{2}+x \right) \\&\quad = (-1)^n C^{(n)} \left( \frac{\pi }{2} -x \right) - (-1)^n C^{(n)} \left( \frac{\pi }{2}+x \right) \\&\quad \ge 0, \\&\frac{d^n}{d x^n} \left( C \left( \frac{\pi }{2} -x \right) + C \left( \frac{\pi }{2} +x \right) \right) \\&\quad = (-1)^n C^{(n)} \left( \frac{\pi }{2} -x \right) + C^{(n)} \left( \frac{\pi }{2}+x \right) \\&\quad = \left\{ (-1)^n C^{(n)} \left( \frac{\pi }{2} -x \right) - (-1)^n C^{(n)} \left( \frac{\pi }{2}+x \right) \right\} +2 C^{(n)} \left( \frac{\pi }{2}+x \right) \\&\quad \ge 0, \quad 0 \le x \le \frac{\pi }{2}, \end{aligned}$$

  and for an odd $n$,

  $$\begin{aligned}&\frac{d^n}{d x^n} \left( C \left( \frac{\pi }{2} -x \right) - C \left( \frac{\pi }{2} +x \right) \right) \\&\quad = \left\{ (-1)^n C^{(n)} \left( \frac{\pi }{2} \!-\!x \right) \!-\! (-1)^n C^{(n)} \left( \frac{\pi }{2}\!+\!x \right) \right\} + 2 (-1)^n C^{(n)} \left( \frac{\pi }{2}+x \right) \ge 0, \\&\frac{d^n}{d x^n} \left( C \left( \frac{\pi }{2} -x \right) + C \left( \frac{\pi }{2} +x \right) \right) \\&\quad = (-1)^n C^{(n)} \left( \frac{\pi }{2} -x \right) - (-1)^n C^{(n)} \left( \frac{\pi }{2}+x \right) \ge 0, \quad 0 \le x \le \frac{\pi }{2}. \end{aligned}$$

  Thus, both $C \left( \frac{\pi }{2} -x \right) - C \left( \frac{\pi }{2} +x \right)$ and $C \left( \frac{\pi }{2} -x \right) + C \left( \frac{\pi }{2} +x \right)$ are absolutely monotone on $\left[ 0, \frac{\pi }{2} \right]$.

Appendix C: Proof of Theorem 3

• (i) It follows from the binomial theorem that

  $$\begin{aligned} C \left( \frac{\pi }{2} -x \right)&= \sum _{k=0}^p b_k \left( \frac{\pi }{2} -x \right) ^k \\&= \sum _{k=0}^p b_k \sum _{j=0}^k (-1)^j {k \atopwithdelims ()j} \left( \frac{\pi }{2} \right) ^{k-j} x^j \\&= \sum _{k=0}^p a_k x^k, \quad x \in \left[ - \frac{\pi }{2}, \frac{\pi }{2} \right] , \end{aligned}$$

  and

  $$\begin{aligned} C \left( \frac{\pi }{2} +x \right)&= \sum _{k=0}^p (-1)^k a_k x^k, \quad x \in \left[ - \frac{\pi }{2}, \frac{\pi }{2} \right] , \end{aligned}$$

  where

  $$\begin{aligned} a_k = (-1)^k \sum _{j=k}^p b_j {j \atopwithdelims ()k} \left( \frac{\pi }{2} \right) ^{j-k}, \quad k =0, 1, \ldots , p. \end{aligned}$$

  Clearly,

  $$\begin{aligned} a_k = \frac{1}{k!} \frac{d^k}{dx^k} C \left( \frac{\pi }{2}- x \right) {\Big |}_{x=0}, \quad k = 1, \ldots , p. \end{aligned}$$

  Under the assumption (7), $a_k$, $a_k+(-1)^k a_k$ and $a_k-(-1)^ka_k$ ($k=0, 1, \ldots , p$) are nonnegative, so that

  $$\begin{aligned} C \left( \frac{\pi }{2} -x \right) + C \left( \frac{\pi }{2} + x \right) = \sum _{k=0}^p (a_k+(-1)^k a_k) x^k \end{aligned}$$

  and

  $$\begin{aligned} C \left( \frac{\pi }{2} -x \right) - C \left( \frac{\pi }{2} + x \right) = \sum _{k=0}^p (a_k-(-1)^k a_k) x^k \end{aligned}$$

  are absolutely monotone on $\left[ 0, \frac{\pi }{2} \right]$. By Theorem 2 (i), $C(\vartheta )$ is a covariance function on $\mathbb {S}^\infty$.

• (ii) Let $C(\vartheta )$ be a covariance function on $\mathbb {S}^\infty$. By Corollary 2,

  $$\begin{aligned} g(x) = C \left( \frac{\pi }{2}- \arcsin x \right) = \sum _{k=0}^p a_k (\arcsin x)^k, \quad x \in [-1, 1], \end{aligned}$$

  is nonnegative, nondecreasing, and convex on $[0, 1]$, so that $g(0) \ge 0,$ $g^{(k)} (0) \ge 0, k =1, 2.$ The first- and second-order derivatives of $g(x)$ are

  $$\begin{aligned} g'(x)&= (1-x^2)^{-\frac{1}{2}} \sum _{k=1}^p k a_k (\arcsin x)^{k-1}, \\ g''(x)&= x (1\!-\!x^2)^{\!-\frac{3}{2}} \!\!\sum _{k=1}^p k a_k (\arcsin x)^{k\!-\!1} \!+\! (1-x^2)^{-1} \sum _{k=2}^p k (k\!-\!1) a_k (\arcsin x)^{k\!-\!2}. \end{aligned}$$

  Thus, inequalities (8) follow from

  $$\begin{aligned} 0&\le g(0) = a_0, \\ 0&\le g'(0) = a_1, \\ 0&\le g''(0) =2 a_2. \end{aligned}$$