Isotropic Covariance Matrix Functions on Compact Two-Point Homogeneous Spaces

Abstract

The covariance matrix function is characterized in this paper for a Gaussian or elliptically contoured vector random field that is stationary, isotropic, and mean square continuous on the compact two-point homogeneous space. Necessary and sufficient conditions are derived for a symmetric and continuous matrix function to be an isotropic covariance matrix function on all compact two-point homogeneous spaces. It is also shown that, for a symmetric and continuous matrix function with compact support, if it makes an isotropic covariance matrix function in the Euclidean space, then it makes an isotropic covariance matrix function on the sphere or the real projective space.Keywords Covariance matrix function · Elliptically contoured random field · Gaussian random field · Isotropy · Stationarity · Jacobi polynomial · Bessel function Mathematics Subject Classification (2010) 60G60 · 62M10 · 62M30 IntroductionA d-dimensional compact two-point homogeneous space M d is a compact Riemannian symmetric space of rank one, and belongs to one of the following five families ([14], [29]): the unit spheres S d (d = 1, 2, . . .), the real projective spaces P d (R) (d = 2, 4, . . .), the complex projective spaces P d (C) (d = 4, 6, . . .), the quaternionic projective spaces P d (H) (d = 8, 12, . . .), and the Cayley elliptic plane P 16 (Cay) or

Strong Local Nondeterminism and Exact Modulus of Continuity for Isotropic Gaussian Random Fields on Compact Two-Point Homogeneous Spaces

“…In what follows let d ≥ 2, α = d−2 2 , and let β be given in the last column of Table 1 associated with α. Our focus is on a Gaussian random field {Z(x), x ∈ M d } that is isotropic and mean square continuous on M d , whose covariance function is known ( [21], [26]) to be of the form (1). This section establishes the property of strong local nondeterminism (SLND) for {Z(x), x ∈ M d } under certain asymptotic condition on the coefficient sequence {b l , l ∈ N 0 } in (1).…”

“…Gaussian random fields on M d have been studied in [4], [9], [14], [21], [26], [28], among others, while theoretical investigations and practical applications of scalar and vector random fields on spheres may be found in [4], [7], [10], [14], [20], [24], [25], [28], [29], [40]- [42]. Recently, a series representation for a real-valued isotropic Gaussian random field on M d is presented in [28,Chapter 2].…”

“…More generally, a series representation is provided in [26] for a vector random field that is isotropic and mean square continuous on M d and stationary on a temporal domain, and a general form of the covariance matrix function is derived for such a vector random field, which involve Jacobi polynomials and the distance defined on M d . Parametric and semiparametric covariance matrix structures on M d are constructed in [21].…”

“…For an isotropic and mean square continuous random field on M d , its covariance function is of the form ( [21], [26])…”

“…) is a function of the form (1), then there exists an isotropic Gaussian or elliptically contoured random field on M d with C(ρ(x 1 , x 2 )) as its covariance function [21], [26].…”