Time-Varying Isotropic Vector Random Fields on Compact Two-Point Homogeneous Spaces

Ma, Chunsheng; Malyarenko, Anatoliy

doi:10.1007/s10959-018-0872-7

Cited by 23 publications

(36 citation statements)

References 36 publications

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“…To prove Theorem 3, we need Lemmas 1 and 2. With identity (25) taken from [1], Lemma 1 (ii) is derived from (25) and Lemma 3 of [21]. The proof of Lemma 2 (ii) is given in Subsection 5.4, while limit (26) in Lemma 2 (i) is from (18.…”

Section: Isotropic Covariance Matrix Functions On All Dimensionsmentioning

confidence: 99%

“…An m-variate isotropic and mean square continuous random field on M d has a series representation [21], for d ≥ 2,…”

mentioning

confidence: 99%

“…On the other hand, there exists an m-variate isotropic Gaussian or elliptically contoured random field on M d with C(ρ(x 1 , x 2 )) as its covariance matrix function [21], if C(ρ(x 1 , x 2 )) is an m × m symmetric matrix function of the form (3). Given a symmetric matrix function C(ϑ ) whose entries are continuous on [0, π], Section 2 presents the characterizations for C(ρ(x 1 , x 2 )) to be the covariance matrix function of an isotropic elliptically contoured vector random field on M d , in terms of the positive definiteness of a sequence of symmetric matrices.…”

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confidence: 99%

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Isotropic Covariance Matrix Functions on Compact Two-Point Homogeneous Spaces

Lü

2019

J Theor Probab

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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

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Section: Isotropic Covariance Matrix Functions On All Dimensionsmentioning

confidence: 99%

“…An m-variate isotropic and mean square continuous random field on M d has a series representation [21], for d ≥ 2,…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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Isotropic Covariance Matrix Functions on Compact Two-Point Homogeneous Spaces

Lü

2019

J Theor Probab

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“…), and the Cayley elliptic plane P 16 (Cay) or P 16 (O). There are at least two different approaches to the subject of compact twopoint homogeneous spaces [26], including an approach based on Lie algebras and a geometric approach, which are used in probabilistic and statistical literature [4], [14], [28], [32]. All compact two-point homogeneous spaces share the same property that all geodesics in a given one of these spaces are closed and have the same length [14].…”

Section: Introductionmentioning

confidence: 99%

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

Lü¹,

Ma²,

Xiao³

2021

Preprint

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“…In the general case, it requires an application of the theory of random sections of vector and tensor bundles over S 2 . Therefore, this paper explores an approach that is different from [33] as outlined below.…”

Section: Introductionmentioning

confidence: 99%

On approximation for fractional stochastic partial differential equations on the sphere

Anh

Broadbridge

Olenko

et al. 2018

Stoch Environ Res Risk Assess

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This paper gives the exact solution in terms of the Karhunen-Loève expansion to a fractional stochastic partial differential equation on the unit sphere S 2 ⊂ R 3 with fractional Brownian motion as driving noise and with random initial condition given by a fractional stochastic Cauchy problem. A numerical approximation to the solution is given by truncating the Karhunen-Loève expansion. We show the convergence rates of the truncation errors in degree and the mean square approximation errors in time. Numerical examples using an isotropic Gaussian random field as initial condition and simulations of evolution of cosmic microwave background (CMB) are given to illustrate the theoretical results.

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Time-Varying Isotropic Vector Random Fields on Compact Two-Point Homogeneous Spaces

Cited by 23 publications

References 36 publications

Isotropic Covariance Matrix Functions on Compact Two-Point Homogeneous Spaces

Isotropic Covariance Matrix Functions on Compact Two-Point Homogeneous Spaces

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

On approximation for fractional stochastic partial differential equations on the sphere

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