Stationary and Isotropic Vector Random Fields on Spheres

Abstract

This paper presents the characterization of the covariance matrix function of a Gaussian or second-order elliptically contoured vector random field on the sphere which is stationary, isotropic, and mean square continuous. This characterization involves an infinite sum of the products of positive definite matrices and Gegenbauer’s polynomials, and may not be available for other non-Gaussian vector random fields on spheres such as a χ ² or log-Gaussian vector random field. We also offer two simple but efficient constructing approaches, and derive some parametric covariance matrix structures on spheres.

Appendix: Proof of Theorem 2

Suppose that C(θ) is the covariance matrix function of an m-variate stationary and isotropic random field $\{ \mathbf {Z}(\mathbf{x}), \mathbf{x}\in \mathbb{S}^{\infty}\}$. Then C _ij (θ)=C _ji (θ),θ∈[0,π], i,j=1,…,m. According to Theorem 2 of Schoenberg (1942), for each k∈{1,…,m}, the direct covariance function C _kk (θ) of a component random field $\{ Z_{k} (\mathbf{x}), \mathbf{x}\in\mathbb{S}^{\infty}\}$ possesses an ultraspherical expansion

(11)

where {b _n (k,k),n=0,1,…} is a summable sequence of nonnegative numbers. Similarly, for i≠j, a scalar random field $\{ Z_{i} (\mathbf {x})+Z_{j} (\mathbf{x} ), \mathbf{x}\in\mathbb{S}^{\infty}\}$ has the covariance function C _ii (θ)+C _jj (θ)+2C _ij (θ) that possesses an ultraspherical expansion

(12)

and a scalar random field $\{ Z_{i} (\mathbf{x})-Z_{j} (\mathbf{x}), \mathbf {x}\in\mathbb{S}^{\infty}\}$ has the covariance function C _ii (θ)+C _jj (θ)−2C _ij (θ) that possesses an ultraspherical expansion

(13)

Taking the difference of Eqs. (12) and (13), we obtain

(14)

where $b_{n} (i, j) = \frac{1}{4} (e^{+}_{n} (i, j)-e^{-}_{n} (i, j)), n = 0, 1, \ldots, i, j =1, 2, \ldots, m$. It follows from Eqs. (11) and (14) that C(θ) adopts an expansion like Eq. (6). Therefore, it remains to show that B _n =(b _n (i,j)) _m×m is positive definite, for each nonnegative integer n. To this end, suppose that a ₁,…,a _m are arbitrary real numbers. Using these constants, we formulate a scalar random field $\{ \sum_{k=1}^{m} a_{k} Z_{k} (\mathbf{x}), \mathbf{x}\in\mathbb {S}^{\infty}\}$. This is a second-order random field on $\mathbb{S}^{\infty}$ with covariance function

where the second equality follows from Eqs. (11) and (14). Applying Theorem 2 of Schoenberg (1942) to the last expansion, we obtain that all its coefficients a′B _n a are nonnegative, that is, each B _n =(b _n (i,j)) _m×m is positive definite.

On the other hand, let C(θ) be an m×m matrix function with expansion of Eq. (6). For each n≥0, it is known (Schoenberg 1942) that cosⁿ θ is positive definite on $\mathbb{S}^{\infty}$, and thus 1cosⁿ θ is a covariance matrix function, where 1 is an m×m matrix with all entries equal 1. By Theorem 6 of Ma (2011b), the Hadamard product of B _n and 1cosⁿ θ, which is the same as B _n cosⁿ θ, is also a covariance matrix function on $\mathbb{S}^{\infty}$. So is C(θ), the sum of the sequence {B _n cosⁿ θ,n=0,1,2,…}.