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

Abstract

A general form of the covariance matrix function is derived in this paper for a vector random field that is isotropic and mean square continuous on a compact connected two-point homogeneous space and stationary on a temporal domain. A series representation is presented for such a vector random field which involves Jacobi polynomials and the distance defined on the compact two-point homogeneous space.

1 Introduction

Consider the sphere $\mathbb {S}^d$ embedded into $\mathbb {R}^{d+1}$ as follows: $\mathbb {S}^d=\{\,\mathbf {x}\in \mathbb {R}^{d+1}:\Vert \mathbf {x}\Vert =1\,\}$, and define the distance between the points $\mathbf {x}_1$ and $\mathbf {x}_2$ by $\rho (\mathbf {x}_1,\mathbf {x}_2)=\cos ^{-1}(\mathbf {x}_1^{\top }\mathbf {x}_2)$. With this distance, any isometry between two pairs of points can be extended to an isometry of $\mathbb {S}^d$. A metric space with such a property is called two-point homogeneous. A complete classification of connected and compact two-point homogeneous spaces is performed in [40]. Besides spheres, the list includes projective spaces over different algebras; see Sect. 2 for details. It turns out that any such space is a manifold. We denote it by $\mathbb {M}^d$, where d is the topological dimension of the manifold. Following [24], denote by $\mathbb {T}$ either the set $\mathbb {R}$ of real numbers or the set $\mathbb {Z}$ of integers, and call it the temporal domain.

Let $(\varOmega ,\mathfrak {F},\mathsf {P})$ be a probability space.

Definition 1

An $\mathbb {R}^m$-valued spatio-temporal random field $\mathbf {Z}(\omega ,\mathbf {x},t):\varOmega \times \mathbb {M}^d \times \mathbb {T}\rightarrow \mathbb {R}^m$ is called (wide-sense) isotropic over $\mathbb {M}^d$ and (wide-sense) stationary over the temporal domain $\mathbb {T}$, if its mean function $\mathsf {E}[\mathbf {Z}(\mathbf {x}; t)]$ equals a constant vector, and its covariance matrix function

$$\begin{aligned} {{\,\mathrm{cov}\,}}(\mathbf {Z}(\mathbf {x}_1; t_1), \mathbf {Z}(\mathbf {x}_2; t_2))= & {} \mathsf {E}\left[ (\mathbf {Z}(\mathbf {x}_1; t_1) -\mathsf {E}[\mathbf {Z}(\mathbf {x}_1; t_1)])(\mathbf {Z}(\mathbf {x}_2; t_2) -\mathsf {E}[\mathbf {Z}(\mathbf {x}_2; t_2)])^{\top }\right] , \\&\mathbf {x}_1, \mathbf {x}_2 \in \mathbb {M}^d, t_1, t_2 \in \mathbb {T}, \end{aligned}$$

depends only on the time lag $t_2-t_1$ between $t_2$ and $t_1$ and the distance $\rho (\mathbf {x}_1,\mathbf {x}_2)$ between $\mathbf {x}_1$ and $\mathbf {x}_2$.

As usual, we omit the argument $\omega \in \varOmega$ in the notation for the random field under consideration. In such a case, the covariance matrix function is denoted by $\mathsf {C} ( \rho (\mathbf {x}_1, \mathbf {x}_2); t)$,

$$\begin{aligned} \mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); t_1-t_2)= & {} \mathsf {E}\left[ (\mathbf {Z}(\mathbf {x}_1; t_1) -\mathsf {E}[\mathbf {Z}(\mathbf {x}_1; t_1)])(\mathbf {Z}(\mathbf {x}_2; t_2) -\mathsf {E}[\mathbf {Z}(\mathbf {x}_2; t_2)])^{\top }\right] , \\&\mathbf {x}_1, \mathbf {x}_2 \in \mathbb {M}^d, t_1, t_2 \in \mathbb {T}. \end{aligned}$$

It is an $m \times m$ matrix function, $\mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); -t) = ( \mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); t) )^{\top }$, and the inequality

$$\begin{aligned} \sum _{i=1}^n \sum _{j=1}^n \mathbf {a}^{\top }_i \mathsf {C} (\rho (\mathbf {x}_i, \mathbf {x}_j); t_i-t_j) \mathbf {a}_j \ge 0 \end{aligned}$$

holds for every $n \in \mathbb {N}$, any $\mathbf {x}_i \in \mathbb {M}^d$, $t_i \in \mathbb {T}$, and $\mathbf {a}_i \in \mathbb {R}^m$ ($i =1, 2, \ldots , n$), where $\mathbb {N}$ stands for the set of positive integers, while $\mathbb {N}_0$ denotes the set of nonnegative integers below. On the other hand, given an $m \times m$ matrix function with these properties, there exists an m-variate Gaussian or elliptically contoured random field $\{\, \mathbf {Z} (\mathbf {x}; t):\mathbf {x} \in \mathbb {M}^d, t \in \mathbb {T}\, \}$ with $\mathsf {C} ( \rho (\mathbf {x}_1, \mathbf {x}_2); t)$ as its covariance matrix function [21].

For a scalar and purely spatial random field $\{\, Z(\mathbf {x}):\mathbf {x} \in \mathbb {M}^d\, \}$ that is isotropic and mean square continuous, its covariance function is continuous and possesses a series representation of the form [8, 14, 37]

$$\begin{aligned} {{\,\mathrm{cov}\,}}( Z (\mathbf {x}_1), Z( \mathbf {x}_2)) = \sum \limits _{n=0}^\infty b_n P_n^{ (\alpha , \beta ) } \left( \cos (\rho (\mathbf {x}_1, \mathbf {x}_2)) \right) ,\quad \mathbf {x}_1, \mathbf {x}_2 \in \mathbb {M}^d, \end{aligned}$$

(1)

where $\{\, b_n:n \in \mathbb {N}_0\, \}$ is a sequence of nonnegative numbers with $\sum \nolimits _{n=0}^\infty b_n P_n^{ (\alpha , \beta ) } (1)$ convergent, $P_n^{ (\alpha , \beta )} (x)$ is a Jacobi polynomial of degree n with a pair of parameters $(\alpha , \beta )$ [1, 38], shown in Table 2. A general form of the covariance matrix function and a series representation are derived in [24] for a vector random field that is isotropic and mean square continuous on a sphere and stationary on a temporal domain. They are extended to $\mathbb {M}^d \times \mathbb {T}$ in this paper.

Isotropic random fields over $\mathbb {S}^d$ with values in $\mathbb {R}^1$ and $\mathbb {C}^1$ were introduced in [35]. Theoretical investigations and practical applications of isotropic scalar-valued random fields on spheres may be found in [7, 11, 12, 19, 43], and vector- and tensor-valued random fields on spheres have been considered in [18, 23, 24, 30], among others. Cosmological applications, in particular, studies of tiny fluctuations of the Cosmic Microwave Background, require development of the theory of random sections of vector and tensor bundles over $\mathbb {S}^2$ [4, 15, 25, 27]. See also surveys of the topic in the monographs [26, 31, 42, 44]. Isotropic random fields on connected compact two-point homogeneous spaces are studied in [2, 14, 28, 29, 33], among others.

Some important properties of $\mathbb {M}^d$, $\rho (\mathbf {x}_1, \mathbf {x}_2)$, and $P_n^{(\alpha , \beta )} (x)$ are reviewed in Sect. 2, and two lemmas are derived: one as a special case of the Funk–Hecke formula on $\mathbb {M}^d$ and the other as a kind of probability interpretation. A series representation is given in Sect. 3 for an isotropic and mean square continuous vector random field on $\mathbb {M}^d$, and a series expression of its covariance matrix function, in terms of Jacobi polynomials. Section 4 deals with a spatio-temporal vector random field on $\mathbb {M}^d\times \mathbb {T}$, which is isotropic and mean square continuous vector random field on $\mathbb {M}^d$ and stationary on $\mathbb {T}$, and obtains a series representation for the random field and a general form for its covariance matrix function. The lemmas and theorems are proved in Appendix A.

2 Compact Two-Point Homogeneous Spaces and Jacobi Polynomials

This section starts by recalling some important properties of the compact connected two-point homogeneous space $\mathbb {M}^d$ and those of Jacobi polynomials and then establishes two useful lemmas on a special case of the Funk–Hecke formula on $\mathbb {M}^d$ and its probability interpretation, which are conjectured in [24]. In what follows, we consider only connected compact two-point homogeneous spaces.

The compact connected two-point homogeneous spaces are shown in the first column of Table 1. Besides spheres, there are projective spaces over the fields $\mathbb {R}$ and $\mathbb {C}$, over the skew field $\mathbb {H}$ of quaternions, and over the algebra $\mathbb {O}$ of octonions. The possible values of d are chosen in such a way that all the spaces in Table 1 are different and exhaust the list. In the lowest dimensions, we have $\mathbb {P}^1(\mathbb {R})=\mathbb {S}^1$, $\mathbb {P}^2(\mathbb {C})=\mathbb {S}^2$, $\mathbb {P}^4(\mathbb {H})=\mathbb {S}^4$, and $\mathbb {P}^8(\mathbb {O})=\mathbb {S}^8$.

Table 1 An approach based on Lie algebras

All compact two-point homogeneous spaces share the same property [6] that all of their geodesic lines are closed. Moreover, all of them are circles and have the same length. In particular, when the sphere $\mathbb {S}^d$ is embedded into the space $\mathbb {R}^{d+1}$ as described in Sect. 1, the length of any geodesic line is equal to that of the unit circle, that is, $2\pi$. It is natural to norm the distance in such a way that the length of any geodesic line is equal to $2\pi$, exactly as in the case of the unit sphere.

There are at least two different approaches to the subject of compact two-point homogeneous spaces in the literature. They are reviewed in the next two subsections.

2.1 An Approach Based on Lie Algebras

This approach goes back to Cartan [10]. It has been used in both the probabilistic literature [14] and the approximation theory literature [3].

Let G be the connected component of the group of isometries of $\mathbb {M}^d$, and let K be the stationary subgroup of a fixed point in $\mathbb {M}^d$, call it $\mathbf {o}$. Cartan [10] defined and calculated the numbers p and q, which are dimensions of some root spaces connected with the Lie algebras of the groups G and K. The groups G and K are listed in the second and the third columns of Table 1, while the numbers p and q are listed in the fourth and fifth columns of the table.

By [17, Theorem 11], if $\mathbb {M}^d$ is a two-point homogeneous space, then the only differential operators on $\mathbb {M}^d$ that are invariant under all isometries of $\mathbb {M}^d$ are the polynomials in a special differential operator $\varDelta$ called the Laplace–Beltrami operator. Let $\hbox {d}\nu (\mathbf {x})$ be the measure which is induced on the homogeneous space $\mathbb {M}^d=G/K$ by the probabilistic invariant measure on G. It is possible to define $\varDelta$ as a self-adjoint operator in the space $H=L^2(\mathbb {M}^d,\hbox {d}\nu (\mathbf {x}))$. The spectrum of $\varDelta$ is discrete, and the eigenvalues are

$$\begin{aligned} \lambda _{n}=-\varepsilon n(\varepsilon n+\alpha +\beta +1), ~~~~~~ n \in \mathbb {N}_0, \end{aligned}$$

where

$$\begin{aligned} \alpha =(p+q-1)/2,\qquad \beta =(q-1)/2, \end{aligned}$$

(2)

and where $\varepsilon =2$ if $\mathbb {M}^d= \mathbb {P}^d(\mathbb {R})$ and $\varepsilon =1$ otherwise.

Let $H_{n}$ be the eigenspace of $\varDelta$ corresponding to $\lambda _{n}$. The space H is the Hilbert direct sum of its subspaces $H_{n}$, $n\in \mathbb {N}_0$. The space $H_n$ is finite-dimensional with

$$\begin{aligned} \dim H_n= \frac{(2n+\alpha +\beta +1)\varGamma (\beta +1) \varGamma (n+\alpha +\beta +1)\varGamma (n+\alpha +1)}{\varGamma (\alpha +1)\varGamma (\alpha +\beta +2)\varGamma (n+1)\varGamma (n+\beta +1)}. \end{aligned}$$

Each of the spaces $H_{n}$ contains a unique one-dimensional subspace whose elements are K-spherical functions; that is, functions invariant under the action of K on $\mathbb {M}^d$. Such a function, say $f_{n}(\mathbf {x})$, depends only on the distance $r=\rho (\mathbf {x},\mathbf {o})$, $f_{n}(\mathbf {x})=f^*_{n}(r)$. A spherical function is called zonal if $f^*_{n}(0)=1$.

The zonal spherical functions of all compact connected two-point homogeneous spaces are listed in the last column of Table 1. To explain notation, we recall that the Jacobi polynomials

$$\begin{aligned} P_n^{(\alpha , \beta )} (x)= & {} \frac{\varGamma (\alpha +n+1)}{n! \varGamma (\alpha +\beta +n+1)}\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) \frac{\varGamma (\alpha +\beta +n+k+1)}{\varGamma ( \alpha +k+1 )} \left( \frac{x-1}{2} \right) ^k,\\ x\in & {} [-1,1],\quad n \in \mathbb {N}_0, \end{aligned}$$

are the eigenfunctions of the Jacobi operator [38, Theorem 4.2.1]

$$\begin{aligned} \varDelta _x=\frac{1}{(1-x)^{\alpha }(1+x)^{\beta }}\frac{\hbox {d}}{\hbox {d}x} \left( (1-x)^{\alpha +1}(1+x)^{\beta +1}\frac{\hbox {d}}{\hbox {d}x}\right) . \end{aligned}$$

In the last column of Table 1, the normalised Jacobi polynomials are introduced,

$$\begin{aligned} R^{(\alpha ,\beta )}_{n}(x)=\frac{P^{(\alpha ,\beta )}_{n}(x)}{P^{(\alpha ,\beta )}_{n}(1)}, \qquad n \in \mathbb {N}_0, \end{aligned}$$

where

$$\begin{aligned} P^{(\alpha ,\beta )}_{n}(1)=\frac{\varGamma (n+\alpha +1)}{\varGamma (n+1)\varGamma (\alpha +1)}. \end{aligned}$$

(3)

The reason for the exceptional behaviour of the real projective spaces is as follows; see [14, 16]. The space $\mathbb {P}^d(\mathbb {R})$ may be constructed by identification of antipodal points on the sphere $\mathbb {S}^d$. An $\hbox {O}(d)$-invariant function f on $\mathbb {P}^d(\mathbb {R})$ can be lifted to an $\hbox {SO}(d)$-invariant function g on $\mathbb {S}^d$ by $g(\mathbf {x})=f(\pi (\mathbf {x}))$, where $\pi$ maps a point $\mathbf {x}\in \mathbb {S}^d$ to the pair of antipodal points $\pi (\mathbf {x})\in \mathbb {P}^d(\mathbb {R})$. This simply means that a function on [0, 1] can be extended to an even function on $[-\,1,1]$. Only the even polynomials can be functions on the so constructed manifold. By [38, Equation (4.1.3)], we have

$$\begin{aligned} P^{(\alpha ,\beta )}_{n}(x)=(-1)^{n}P^{(\beta ,\alpha )}_{n}(-x). \end{aligned}$$

For the real projective spaces $\alpha =\beta$, and the corresponding normalised Jacobi polynomials are even if and only if n is even.

Remark 1

If two Lie groups have the same connected component of identity, then they have the same Lie algebra. For example, the groups $\hbox {SO}(d)$ and $\hbox {O}(d)$ have the same Lie algebra $\mathfrak {so}(d)$. That is, the approach based on Lie algebras gives the same values of p and q for spheres and real projective spaces of equal dimensions. Only zonal spherical functions can distinguish between the two cases.

In the only case of $\mathbb {M}^d=\mathbb {S}^1$, we have $p=q=0$. The reason is that only in this case the Lie algebra $\mathfrak {so}(2)$ is commutative rather than semisimple, and does not have nonzero root spaces at all.

2.2 A Geometric Approach

There is a trick that allows us to write down all zonal spherical functions of all compact two-point homogeneous spaces in the same form, which is used in probabilistic literature [2, 26, 28, 29, 33] and in approximation theory [9, 13]. Denote $y=\cos (\rho (\mathbf {x},\mathbf {o})/2)$. Then we have $\cos (\rho (\mathbf {x},\mathbf {o}))=2y^2-1$. For the case of $\mathbb {M}^d= \mathbb {P}^d(\mathbb {R})$, $\alpha =\beta =(d-2)/2$. By [38, Theorem 4.1],

$$\begin{aligned} P^{(\alpha ,\alpha )}_{2n}(y)=\frac{\varGamma (2n+\alpha +1)\varGamma (n+1)}{\varGamma (n+\alpha +1)\varGamma (2 n+1)}P^{(\alpha ,-1/2)}_{n}(2y^2-1). \end{aligned}$$

In terms of the normalised Jacobi polynomials, we obtain

$$\begin{aligned} R^{(\alpha ,\alpha )}_{2n}(\cos (\rho (\mathbf {x},\mathbf {o})/2)) =R^{(\alpha ,-1/2)}_{n}(\cos (\rho (\mathbf {x},\mathbf {o}))). \end{aligned}$$

For the case of $\mathbb {M}^d= \mathbb {P}^d(\mathbb {R})$, if we redefine $\alpha =(d-2)/2$, $\beta =-1/2$, then all zonal spherical functions of all compact two-point homogeneous spaces are given by the same expression $R^{(\alpha ,\beta )}_{n}(\cos (\rho (\mathbf {x},\mathbf {o})))$.

It easily follows from (2) that the new values for p and q in the case of $\mathbb {M}^d=P^d(\mathbb {R})$ are $p=d-1$ and $q=0$. It is interesting to note that the new values of p and q for the real projective spaces together with their old values for the rest of spaces still have a meaning; see [13] and Table 2. This time, the values of p and q are connected with the geometry of the space $\mathbb {M}^d$ rather than with Lie algebras.

Table 2 A geometric approach

Specifically, let $\mathbb {A}=\{\,\mathbf {x}\in \mathbb {M}^d :\rho (\mathbf {x},\mathbf {o})=\pi \,\}$. This set is called the antipodal manifold of the point $\mathbf {o}$. The antipodal manifolds are listed in the sixth column of Table 2. Geometrically, if $\mathbb {M}^d=\mathbb {S}^d$ and $\mathbf {o}$ is the North pole, then $\mathbb {A}=\mathbb {S}^0$ is the South pole. Otherwise, $\mathbb {A}$ is the space at infinity of the point $\mathbf {o}$ in the terms of projective geometry. The new number p turns out to be the dimension of the antipodal manifold, while the number $p+q+1$ is, as before, the dimension of the space $\mathbb {M}^d$ itself.

In what follows, we use the geometric approach. It turns out that all the spaces $\mathbb {M}^d$ are Riemannian manifolds, as is defined in [5]. Each Riemannian manifold carries the canonical measure$\mu$; see [5, pp. 10–11]. The measure $\mu$ is proportional to the measure $\nu$ constructed in Sect. 2.1. The coefficient of proportionality or the total measure $\mu (\mathbb {M}^d)$ of the compact manifold $\mathbb {M}^d$ is called the volume of $\mathbb {M}^d$.

Lemma 1

The volume of the space $\mathbb {M}^d$ is

$$\begin{aligned} \omega _d=\mu (\mathbb {M}^d)=\frac{(4\pi )^{\alpha +1} \varGamma (\beta +1)}{\varGamma (\alpha +\beta +2)}. \end{aligned}$$

(4)

In what follows, we write just $\hbox {d}\mathbf {x}$ instead of $\hbox {d}\mu (\mathbf {x})$.

2.3 Orthogonal Properties of Jacobi Polynomials

The set of Jacobi polynomials $\{\, P_n^{(\alpha , \beta )} (x):n \in \mathbb {N}_0, x \in \mathbb {R}\, \}$ possesses two types of orthogonal properties. First, for each pair of $\alpha >-1$ and $\beta >-1$, this set is a complete orthogonal system on the interval $[-\,1, 1]$ with respect to the weight function $(1-x)^\alpha (1+x)^\beta$, in the sense that

$$\begin{aligned} \int _{-1}^1 P^{(\alpha , \beta )}_i (x) P^{(\alpha , \beta )}_j (x) (1-x)^\alpha (1+x)^\beta \hbox {d}x = \left\{ \begin{array}{ll} \frac{2^{\alpha +\beta +1} }{2 j +\alpha +\beta +1} \frac{\varGamma (j+\alpha +1) \varGamma (j+\beta +1)}{ j! \varGamma ( j +\alpha +\beta +1) }, ~ &{} ~ i =j, \\ 0, ~ &{} ~ i \ne j. \end{array}\right. \end{aligned}$$

(5)

Second, for selected values of $\alpha$ and $\beta$ given by (2) with p and q given in Table 2, they are orthogonal over $\mathbb {M}^d$, as the following lemma describes, which is derived from the Funk–Hecke formula recently established in [3]. In the particular case $\mathbb {M}^d=\mathbb {S}^d$, the Funk–Hecke formula may be found in classical references such as [1, 34].

Lemma 2

For $i, j \in \mathbb {N}_0$, and $\mathbf {x}_1$, $\mathbf {x}_2 \in \mathbb {M}^d$,

$$\begin{aligned} \int _{\mathbb {M}^d } P_i^{(\alpha ,\beta ) } (\cos (\rho (\mathbf {x}_1,\mathbf {x}))) P_j^{(\alpha ,\beta )} (\cos (\rho (\mathbf {x}_2,\mathbf {x})))\,\mathrm{d}\mathbf {x} =\frac{\delta _{ij}\omega _d}{a_i^2} P_i^{(\alpha ,\beta )} (\cos (\rho (\mathbf {x}_1,\mathbf {x}_2))), \end{aligned}$$

where

$$\begin{aligned} a_n=\left( \frac{\varGamma (\beta +1)(2 n +\alpha +\beta +1)\varGamma (n+\alpha +\beta +1)}{\varGamma (\alpha +\beta +2)\varGamma (n+\beta +1)}\right) ^{\frac{1}{2}},\qquad n \in \mathbb {N}_0. \end{aligned}$$

(6)

The probabilistic interpretation of zonal spherical functions on $\mathbb {M}^d$ is provided in Lemma 3. The spherical case is given in [23].

Definition 2

A random vector $\mathbf {U}$ is said to be uniformly distributed on $\mathbb {M}^d$ if, for every Borel set $A\subseteq \mathbb {M}^d$ and every isometry g we have $\mathsf {P} (\mathbf {U}\in A ) =\mathsf {P} (\mathbf {U}\in gA)$.

To construct $\mathbf {U}$, we start with a measure $\sigma$ proportional to the invariant measure $\nu$ of Sect. 2.1. Let $T_{\mathbf {o}}$ be the tangent space to $\mathbb {M}^d$ at the point $\mathbf {o}$. Choose a Cartesian coordinate system in $T_{\mathbf {o}}$ and identify this space with the space $\mathbb {R}^{d}$. Construct a chart $\varphi :\mathbb {M}^d\setminus \mathbb {A}\rightarrow \mathbb {R}^{d}$ as follows. Put $\varphi (\mathbf {o})=\mathbf {0}\in \mathbb {R}^d$. For any other point $\mathbf {x}\in \mathbb {M}^d\setminus \mathbb {A}$, draw the unique geodesic line connecting $\mathbf {o}$ and $\mathbf {x}$. Let $\mathbf {r}\in \mathbb {R}^{d}$ be the unit tangent vector to the above geodesic line. Define

$$\begin{aligned} \varphi (\mathbf {x})= \mathbf {r} \tan (\rho (\mathbf {x},\mathbf {o})/2), \end{aligned}$$

and, for each Borel set $B\subseteq \mathbb {M}^d$,

$$\begin{aligned} \sigma (B)=\int _{\varphi ^{-1}(B\setminus \mathbb {A})}\frac{\hbox {d}\mathbf {x}}{(1+\Vert \mathbf {x}\Vert ^2)^{\alpha +\beta +2}}. \end{aligned}$$

This measure is indeed invariant [39, p. 113]. Finally, define a probability space $(\varOmega ',$$\mathfrak {F}',$$\mathsf {P}')$ as follows: $\varOmega '=\mathbb {M}^d$, $\mathfrak {F}'$ is the $\sigma$-field of Borel subsets of $\varOmega '$, and

$$\begin{aligned} \mathsf {P}'(B)=\frac{\sigma (B)}{\sigma (\mathbb {M}^d)},\qquad B\in \mathfrak {B}'. \end{aligned}$$

The random variable $\mathbf {U}(\omega )=\omega$ is then uniformly distributed on $\mathbb {M}^d$.

Lemma 3

Let $\mathbf {U}$ be a random vector uniformly distributed on $\mathbb {M}^d$. For $n \in \mathbb {N}$,

$$\begin{aligned} Z_n(\mathbf {x})=a_n P_n^{(\alpha ,\beta )} (\cos (\rho (\mathbf {x},\mathbf {U}))), \qquad \mathbf {x}\in \mathbb {M}^d, \end{aligned}$$

is a centred isotropic random field with covariance function

$$\begin{aligned} {{\,\mathrm{cov}\,}}( Z_n (\mathbf {x}_1), Z_n (\mathbf {x}_2) ) =P_n^{ (\alpha ,\beta )} (\cos (\rho (\mathbf {x}_1, \mathbf {x}_2))), ~~~~~ \mathbf {x}_1, \mathbf {x}_2 \in \mathbb {M}^d, \end{aligned}$$

where $a_n$ is given by (6). Moreover, for $k \ne n$, the random fields $\{\, Z_k (\mathbf {x}):\mathbf {x} \in \mathbb {M}^d\, \}$ and $\{\,Z_n(\mathbf {x}): \mathbf {x} \in \mathbb {M}^d\, \}$ are uncorrelated:

$$\begin{aligned} {{\,\mathrm{cov}\,}}(Z_k (\mathbf {x}_1), Z_n (\mathbf {x}_2) ) =0, ~~~~ \mathbf {x}_1, \mathbf {x}_2 \in \mathbb {M}^d. \end{aligned}$$

(7)

3 Isotropic Vector Random Fields on $\mathbb {M}^d$

In the purely spatial case, this section presents a series representation for an m-variate isotropic and mean square continuous random field $\{\, \mathbf {Z} (\mathbf {x}):\mathbf {x} \in \mathbb {M}^d\, \}$ and a series expression for its covariance matrix function, in terms of Jacobi polynomials. By mean square continuous, we mean that, for $k =1, \ldots , m$,

$$\begin{aligned} \mathsf {E}\left[ | Z_k (\mathbf {x}_1) -Z_k (\mathbf {x}_2) |^2\right] \rightarrow 0, ~~ \text{ as } ~~ \rho (\mathbf {x}_1, \mathbf {x}_2 ) \rightarrow 0, ~ \mathbf {x}_1, \mathbf {x}_2 \in \mathbb {M}^d. \end{aligned}$$

It implies the continuity of each entry of the associated covariance matrix function in terms of $\rho (\mathbf {x}_1, \mathbf {x}_2)$.

In what follows, d is assumed to be greater than 1, while $\mathbb {M}^d$ reduces to the unit circle $\mathbb {S}^1$ when $d=1$, over which the treatment of isotropic vector random fields may be found in [23, 24]. For an $m \times m$ symmetric and nonnegative definite matrix $\mathsf {B}$ with nonnegative eigenvalues $\lambda _1, \dots$, $\lambda _m$, there is an orthogonal matrix $\mathsf {S}$ such that $\mathsf {S}^{-1}\mathsf {B}\mathsf {S}=\mathsf {D}$, where $\mathsf {D}$ is a diagonal matrix with diagonal entries $\lambda _1, \ldots , \lambda _m$. Define the square root of $\mathsf {B}$ by

$$\begin{aligned} \mathsf {B}^{\frac{1}{2}}=\mathsf {S}\mathsf {D}^{\frac{1}{2}}\mathsf {S}^{-1}, \end{aligned}$$

where $\mathsf {D}^{\frac{1}{2}}$ is a diagonal matrix with diagonal entries $\sqrt{\lambda _1}, \ldots , \sqrt{ \lambda _m}$. Clearly, $\mathsf {B}^{\frac{1}{2}}$ is symmetric, nonnegative definite, and $(\mathsf {B}^{\frac{1}{2}})^2=\mathsf {B}$. Denote by $\mathsf {I}_m$ an $m \times m$ identity matrix. For a sequence of $m \times m$ matrices $\{\, \mathsf {B}_n:n \in \mathbb {N}_0 \,\}$, the series $\sum \nolimits _{n=0}^\infty \mathsf {B}_n$ is said to be convergent, if each of its entries is convergent.

Theorem 1

Suppose that $\{\, \mathbf {V}_n:n \in \mathbb {N}_0\, \}$ is a sequence of independent m-variate random vectors with $\mathsf {E} ( \mathbf {V}_n)= \mathbf {0}$ and ${{\,\mathrm{cov}\,}}( \mathbf {V}_n, \mathbf {V}_n ) = a_n^2\mathsf {I}_m$, $\mathbf {U}$ is a random vector uniformly distributed on $\mathbb {M}^d$ and is independent of $\{\, \mathbf {V}_n:n \in \mathbb {N}_0\, \}$, and that $\{\, \mathsf {B}_n:n \in \mathbb {N}_0\, \}$ is a sequence of $m \times m$ symmetric nonnegative definite matrices. If the series $\sum \nolimits _{n=0}^\infty \mathsf {B}_n P_n^{ (\alpha , \beta ) } (1)$ converges, then

$$\begin{aligned} \mathbf {Z} (\mathbf {x}) = \sum _{n=0}^\infty \mathsf {B}_n^{\frac{1}{2}} \mathbf {V}_n P_n^{ (\alpha , \beta ) } ( \cos \rho (\mathbf {x}, \mathbf {U} )), ~~~~~~ \mathbf {x} \in \mathbb {M}^d, \end{aligned}$$

(8)

is a centred m-variate isotropic random field on $\mathbb {M}^d$, with covariance matrix function

$$\begin{aligned} {{\,\mathrm{cov}\,}}( \mathbf {Z} (\mathbf {x}_1), \mathbf {Z}(\mathbf {x}_2) ) = \sum _{n=0}^\infty \mathsf {B}_n P_n^{(\alpha , \beta ) } \left( \cos \rho (\mathbf {x}_1, \mathbf {x}_2) \right) , ~~~~~~ \mathbf {x}_1, \mathbf {x}_2 \in \mathbb {M}^d. \end{aligned}$$

(9)

The terms of (8) are uncorrelated; more precisely,

$$\begin{aligned} {{\,\mathrm{cov}\,}}\left( \mathsf {B}_i^{\frac{1}{2}} \mathbf {V}_i P_i^{ (\alpha , \beta ) } ( \rho (\mathbf {x}_1, \mathbf {U})), ~ \mathsf {B}_j^{\frac{1}{2}} \mathbf {V}_j P_j^{ (\alpha , \beta ) } ( \rho (\mathbf {x}_2, \mathbf {U} )) \right) = \mathbf {0}, ~~~ \mathbf {x}_1, \mathbf {x}_2 \in \mathbb {M}^d, ~ i \ne j. \end{aligned}$$

Since $\left| P_n^{ (\alpha , \beta ) } (\cos \vartheta ) \right| \le P_n^{ (\alpha , \beta ) } (1), n \in \mathbb {N}_0,$ the convergent assumption of the series $\sum \nolimits _{n=0}^\infty \mathsf {B}_n P_n^{ (\alpha , \beta ) } (1)$ ensures not only the mean square convergence of the series at the right-hand side of (8), but also the uniform and absolute convergence of the series at the right-hand side of (9).

When $\mathbb {M}^d=\mathbb {S}^2$ and $m=1$, we have $\dim H_n=2n+1$, and (9) takes the form

$$\begin{aligned} {{\,\mathrm{cov}\,}}( Z (\mathbf {x}_1), Z(\mathbf {x}_2) ) = \sum _{n=0}^\infty b_n P_n\left( \cos \rho (\mathbf {x}_1, \mathbf {x}_2) \right) , \end{aligned}$$

where $P_n (x)$ are Legendre polynomials. In the theory of Cosmic Microwave Background, this equation is traditionally written in the form

$$\begin{aligned} {{\,\mathrm{cov}\,}}( Z (\mathbf {x}_1), Z(\mathbf {x}_2) ) = \sum _{\ell =0}^\infty (2\ell +1)C_{\ell } P_{\ell }\left( \mathbf {x}_1\cdot \mathbf {x}_2\right) , \end{aligned}$$

and the sequence $\{\,C_{\ell }:\ell \ge 0\,\}$ is called the angular power spectrum. In the general case, define the angular power spectrum by

$$\begin{aligned} \mathsf {C}_n=\frac{1}{\dim H_n}\mathsf {B}_n. \end{aligned}$$

A lot of examples of the angular power spectrum for general compact two-point homogeneous spaces may be found in [2].

As the next theorem indicates, (9) is a general form that the covariance matrix function of an m-variate isotropic and mean square continuous random field on $\mathbb {M}^d$ must take.

Theorem 2

For an m-variate isotropic and mean square continuous random field $\{\, Z(\mathbf {x}):\mathbf {x} \in \mathbb {M}^d\, \}$, its covariance matrix function ${{\,\mathrm{cov}\,}}( Z(\mathbf {x}_1), Z (\mathbf {x}_2) )$ is of the form

$$\begin{aligned} \mathsf {C} ( \mathbf {x}_1, \mathbf {x}_2 ) = \sum _{n=0}^\infty \mathsf {B}_n P_n^{ (\alpha , \beta ) } \left( \cos \rho (\mathbf {x}_1, \mathbf {x}_2) \right) , ~~~~~~ \mathbf {x}_1, \mathbf {x}_2 \in \mathbb {M}^d, \end{aligned}$$

(10)

where $\{\,\mathsf {B}_n:n \in \mathbb {N}_0\, \}$ is a sequence of $m \times m$ nonnegative definite matrices and the series $\sum \nolimits _{n=0}^\infty \mathsf {B}_n P_n^{ (\alpha , \beta ) } (1)$ converges.

Conversely, if an $m \times m$ matrix function $\mathsf {C} (\mathbf {x}_1, \mathbf {x}_2)$ is of the form (10), then it is the covariance matrix function of an m-variate isotropic Gaussian or elliptically contoured random field on $\mathbb {M}^d$.

Examples of covariance matrix functions on $\mathbb {S}^d$ may be found in, for instance, [23, 24]. We would call for parametric and semi-parametric covariance matrix structures on $\mathbb {M}^d$.

4 Time-Varying Isotropic Vector Random Fields on $\mathbb {M}^d$

For an m-variate random field $\{\, \mathbf {Z} (\mathbf {x}; t):\mathbf {x} \in \mathbb {M}^d, t \in \mathbb {T}\, \}$ that is isotropic and mean square continuous over $\mathbb {M}^d$ and stationary on $\mathbb {T}$, this section presents the general form of its covariance matrix function $\mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); t)$, which is a continuous function of $\rho (\mathbf {x}_1, \mathbf {x}_2)$ and is also a continuous function of $t \in \mathbb {R}$ if $\mathbb {T} = \mathbb {R}$. A series representation is given in the following theorem for such a random field, as an extension of that on $\mathbb {S}^d \times \mathbb {T}$.

Theorem 3

If an m-variate random field $\{ \mathbf {Z} (\mathbf {x}; t), \mathbf {x} \in \mathbb {M}^d, t \in \mathbb {T} \}$ is isotropic and mean square continuous over $\mathbb {M}^d$ and stationary on $\mathbb {T}$, then

$$\begin{aligned} \mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); -t) = ( \mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); t) )^{\top }, \end{aligned}$$

and $\frac{\mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); t) + \mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); -t)}{2}$ is of the form

$$\begin{aligned}&\frac{\mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); t) + \mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); -t)}{2} \nonumber \\&\quad = \sum \limits _{n=0}^\infty \mathsf {B}_n (t) P_n^{ (\alpha , \beta ) } (\cos \rho (\mathbf {x}_1, \mathbf {x}_2)), \quad \mathbf {x}_1, \mathbf {x}_2\in \mathbb {M}^d, t\in \mathbb {T}, \end{aligned}$$

(11)

where, for each fixed $n \in \mathbb {N}_0$, $\mathsf {B}_n (t)$ is a stationary covariance matrix function on $\mathbb {T}$, and, for each fixed $t \in \mathbb {T}$, $\mathsf {B}_n (t)$ ($n \in \mathbb {N}_0$) are $m \times m$ symmetric matrices and $\sum \nolimits _{n=0}^\infty \mathsf {B}_n (t) P_n^{ (\alpha , \beta ) } (1)$ converges.

While a general form of $\frac{\mathsf {C} ( \rho (\mathbf {x}_1, \mathbf {x}_2); t) + \mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); -t)}{2}$, instead of $\mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); t)$ itself, is given in Theorem 3, that of $\mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); t)$ can be obtained in certain special cases, such as spatio-temporal symmetric, and purely spatial.

Corollary 1

If $\mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); t)$ is spatio-temporal symmetric in the sense that

$$\begin{aligned} \mathsf {C} ( \rho (\mathbf {x}_1, \mathbf {x}_2); - t ) =\mathsf {C} ( \rho (\mathbf {x}_1, \mathbf {x}_2); t ), ~~~~~~~~ \mathbf {x}_1, \mathbf {x}_2 \in \mathbb {M}^d, ~ t \in \mathbb {T}, \end{aligned}$$

then it takes the form

$$\begin{aligned} \mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); t) = \sum \limits _{n=0}^\infty \mathsf {B}_n (t) P_n^{ (\alpha , \beta ) } (\cos \rho (\mathbf {x}_1, \mathbf {x}_2)), ~~ \mathbf {x}_1, \mathbf {x}_2 \in \mathbb {M}^d, ~ t \in \mathbb {T}. \end{aligned}$$

In contrast to those in (11), the $m \times m$ matrices $\mathsf {B}_n (t)$ ($n \in \mathbb {N}_0$) in the next theorem are not necessarily symmetric. One simple such example is

which is the covariance matrix function of an m-variate first order moving average time series , where $\{\, \varvec{\varepsilon } (t):t \in \mathbb {Z}\, \}$ is m-variate white noise with $\mathsf {E}[ \varvec{\varepsilon } (t)] = \mathbf {0}$ and , and is an $m \times m$ matrix.

Theorem 4

An $m \times m$ matrix function

$$\begin{aligned} \mathsf {C} ( \rho (\mathbf {x}_1, \mathbf {x}_2); t) = \sum \limits _{n=0}^\infty \mathsf {B}_n (t) P_n^{ (\alpha , \beta ) } (\cos \rho (\mathbf {x}_1, \mathbf {x}_2)), ~ ~~ \mathbf {x}_1, \mathbf {x}_2 \in \mathbb {M}^d, ~ t \in \mathbb {T}, \end{aligned}$$

(12)

is the covariance matrix function of an m-variate Gaussian or elliptically contoured random field on $\mathbb {M}^d \times \mathbb {T}$ if and only if $\{\, \mathsf {B}_n (t):n \in \mathbb {N}_0\, \}$ is a sequence of stationary covariance matrix functions on $\mathbb {T}$ and $\sum \nolimits _{n=0}^\infty \mathsf {B}_n (0) P_n^{ (\alpha , \beta ) } (1)$ converges.

As an example of (12), let

where is a sequence of $m \times m$ nonnegative definite matrices and $P_n^{ (\alpha , \beta ) } (1)$ converges. In this case, (12) is the covariance matrix function of an m-variate Gaussian or elliptically contoured random field on $\mathbb {M}^d \times \mathbb {Z}$.

Gaussian and second-order elliptically contoured random fields form one of the largest sets, if not the largest set, which allows any possible correlation structure [21]. The covariance matrix functions developed in Theorem 4 can be adopted for a Gaussian or elliptically contoured vector random field. However, they may not be available for other non-Gaussian random fields, such as a log-Gaussian [32], $\chi ^2$ [20], K-distributed [22], or skew-Gaussian one, for which admissible correlation structure must be investigated on a case-by-case basis. A series representation is given in the following theorem for an m-variate spatio-temporal random field on $\mathbb {M}^d\times \mathbb {T}$.

Theorem 5

An m-variate random field

$$\begin{aligned} \mathbf {Z} (\mathbf {x}; t) = \sum _{n=0}^\infty \mathbf {V}_n (t) P_n^{ (\alpha , \beta ) } ( \cos \rho (\mathbf {x}, \mathbf {U})), ~~~~~~ \mathbf {x} \in \mathbb {M}^d, ~ t \in \mathbb {T}, \end{aligned}$$

(13)

is isotropic and mean square continuous on $\mathbb {M}^d$, stationary on $\mathbb {T}$, and possesses mean $\mathbf {0}$ and covariance matrix function (12), where $\{ \,\mathbf {V}_n (t):n \in \mathbb {N}_0 \, \}$ is a sequence of independent m-variate stationary stochastic processes on $\mathbb {T}$ with

$$\begin{aligned} \mathsf {E} ( \mathbf {V}_n )= \mathbf {0}, ~~~ {{\,\mathrm{cov}\,}}( \mathbf {V}_n (t_1), \mathbf {V}_n (t_2) ) = a_n^2 \mathsf {B}_n (t_1-t_2), ~~~ n \in \mathbb {N}_0, \end{aligned}$$

the random vector $\mathbf {U}$ is uniformly distributed on $\mathbb {M}^d$ and is independent with $\{\, \mathbf {V}_n (t) :$$n \in \mathbb {N}_0\, \}$, and $\sum \nolimits _{n=0}^\infty \mathsf {B}_n (0) P_n^{ (\alpha , \beta ) } (1)$ converges.

The distinct terms of (13) are uncorrelated each other,

$$\begin{aligned}&{{\,\mathrm{cov}\,}}\left( \mathbf {V}_i (t) P_i^{ (\alpha , \beta ) } ( \cos \rho (\mathbf {x}, \mathbf {U}) ), ~ \mathbf {V}_j (t) P_j^{ (\alpha , \beta ) } ( \cos \rho (\mathbf {x}, \mathbf {U}) ) \right) = \mathbf {0},\\&\quad \mathbf {x} \in \mathbb {M}^d, ~ t \in \mathbb {T}, i \ne j, \end{aligned}$$

due to Lemma 3 and the independent assumption among $\mathbf {U}, \mathbf {V}_i (t), \mathbf {V}_j (t)$. The vector stochastic process $\mathbf {V}_n (t)$ can be expressed as, in terms of $\mathbf {Z} (\mathbf {x}; t)$ and $\mathbf {U}$,

$$\begin{aligned} \mathbf {V}_n (t) = \frac{a^2_n}{\omega _d P_n^{ (\alpha , \beta ) } (1)} \int _{\mathbb {M}^d} \mathbf {Z} (\mathbf {x}; t) P_n^{ (\alpha , \beta ) } (\cos \rho (\mathbf {x}, \mathbf {U})) \mathrm{d} \mathbf {x}, ~~~~~ t \in \mathbb {T}, ~ n \in \mathbb {N}_0, \end{aligned}$$

where the integral is understood as a Bochner integral of a function taking values in the Hilbert space of random vectors $\mathbf {Z}\in \mathbb {R}^m$ with $\mathsf {E}[\Vert \mathbf {Z}\Vert ^2_{\mathbb {R}^m}]<\infty$.

It is obtained after we multiply both sides of (13) by $P_n^{ (\alpha , \beta ) } (\cos \rho (\mathbf {x}, \mathbf {U}))$, integrate over $\mathbb {M}^d$, and apply Lemma 3,

$$\begin{aligned}&\int _{\mathbb {M}^d} \mathbf {Z} (\mathbf {x}; t) P_n^{ (\alpha , \beta ) } (\cos \rho (\mathbf {x}, \mathbf {U})) \mathrm{d} \mathbf {x}\\&\quad = \sum _{k=0}^\infty \mathbf {V}_n (t) \int _{\mathbb {M}^d} P_k^{ (\alpha , \beta ) } ( \cos \rho (\mathbf {x}, \mathbf {U}) ) P_n^{(\alpha , \beta ) } ( \cos \rho (\mathbf {x}, \mathbf {U})) \mathrm{d} \mathbf {x} \\&\quad = \frac{1}{a_n^2} P_n^{ (\alpha , \beta ) } (1) \mathbf {V}_n (t). \end{aligned}$$

A Proofs

Proof of Lemma 1

To calculate $\mu (\mathbb {M}^d)$, we use the result of [41]. If all the geodesics on a d-dimensional Riemannian manifold M are closed and have length $2\pi L$, then the ratio

$$\begin{aligned} i(M)=\frac{\mu (\mathbb {M}^d)}{L^n\mu (\mathbb {S}^d)} \end{aligned}$$

is an integer. With our convention $L=1$, we obtain $\mu (\mathbb {M}^d)=i(\mathbb {M}^d)\mu (\mathbb {S}^d)$. It is well known that

$$\begin{aligned} \mu (\mathbb {S}^d)=\frac{2\pi ^{(d+1)/2}}{\varGamma ((d+1)/2)} =\frac{2\pi ^{\alpha +3/2}}{\varGamma (\alpha +3/2)}. \end{aligned}$$

(14)

The Weinstein’s integers$i(\mathbb {M}^d)$ are shown in the last column of Table 2. Following [36], consider all the geodesics from $\mathbf {o}$ to a point in $\mathbb {A}$. Draw a tangent line to each of them and denote by e the dimension of the linear space generated by these lines. We have $e=d$ for $\mathbb {S}^d$, 1 for $P^d(\mathbb {R})$, 2 for $P^d(\mathbb {C})$, 4 for $P^d(\mathbb {H})$, and 8 for $P^2(\mathbb {O})$. It is proved in [36] that

$$\begin{aligned} i(\mathbb {M}^d)=\frac{2^{d-1}\varGamma ((d+1)/2)\varGamma (e/2)}{\sqrt{\pi }\varGamma ((d+e)/2)} \end{aligned}$$

We know that $d=2\alpha +2$. It is easy to check that $e=2\beta +2$, then we obtain

$$\begin{aligned} i(\mathbb {M}^d)=\frac{2^{2\alpha +1}\varGamma (\alpha +3/2) \varGamma (\beta +1)}{\sqrt{\pi }\varGamma (\alpha +\beta +2)}, \end{aligned}$$

and (4) easily follows. $\square$

Proof of Lemma 2

In Theorem 2.1 of [3], put $K(x)=P_i^{ (\alpha ,\beta )} (x)$ and $S(\mathbf {x})=P_j^{(\alpha ,\beta )} (\cos (\rho (\mathbf {x}_2,\mathbf {x})))$. We obtain

$$\begin{aligned}&\int _{\mathbb {M}^d}P_i^{ (\alpha ,\beta )} (\cos (\rho (\mathbf {x}_1,\mathbf {x}))) P_j^{ (\alpha ,\beta )} (\cos (\rho (\mathbf {x}_2,\mathbf {x})))\,\hbox {d}\mathbf {x} \\&\quad = \omega _d P_j ^{(\alpha ,\beta )} (\cos (\rho (\mathbf {x}_1,\mathbf {x}_2))) \int _{-1}^{1} \frac{P_i^{(\alpha ,\beta )} (x)}{ P_i^{(\alpha ,\beta )} (1)} P_j^{(\alpha ,\beta )} (x) \hbox {d}\nu _{\alpha ,\beta }(x) \\&\quad = \omega _d \frac{\delta _{ij}}{a_i^2} P_i^{(\alpha ,\beta )} (\cos (\rho (\mathbf {x}_1,\mathbf {x}_2))), \end{aligned}$$

where the last equality follows from (3) , (5), and the following well-known result: the probabilistic measure $\nu _{\alpha ,\beta }$ on $[-\,1,1]$, proportional to $(1-x)^{\alpha }(1+x)^{\beta }\,\hbox {d}x$, is

$$\begin{aligned} \hbox {d}\nu _{\alpha ,\beta }(x)=\frac{\varGamma (\alpha +\beta +2)}{2^{\alpha +\beta +1} \varGamma (\alpha +1)\varGamma (\beta +1)}(1-x)^{\alpha } (1+x)^{\beta }\,\hbox {d}x. \end{aligned}$$

(15)

$\square$

Proof of Lemma 3

The mean function of $\{\, Z_n (\mathbf {x}):\mathbf {x} \in \mathbb {M}^d\, \}$ is obtained by applying of [3, Theorem 2.1] to $K(x)=1$ and $S(\mathbf {x})=P^{(\alpha ,\beta )}_n (\cos (\rho (\mathbf {x},\mathbf {y})))$,

$$\begin{aligned} \mathsf {E}[Z_n (\mathbf {x})] = a_n\omega _d \int _{\mathbb {M}^d} P_n^{(\alpha ,\beta )} (\cos (\rho (\mathbf {x},\mathbf {y}))) \,\hbox {d}\mathbf {y} = a_n \cdot 0 = 0. \end{aligned}$$

The covariance function is

$$\begin{aligned} {{\,\mathrm{cov}\,}}( Z_n (\mathbf {x}_1), Z_n(\mathbf {x}_2) )= & {} \omega _d^{-1}a_n^2\int _{\mathbb {M}^d} P_n^{ (\alpha ,\beta )} (\cos (\rho ( \mathbf {x}_1, \mathbf {z})) P_n^{ (\alpha ,\beta )} (\cos (\rho (\mathbf {x}_2,\mathbf {z})))\,\hbox {d}\mathbf {z}\\= & {} P_n^{(\alpha ,\beta )} (\cos (\rho (\mathbf {x}_1, \mathbf {x}_2) ), \end{aligned}$$

by Lemma 2. Equation (7) easily follows from the same lemma. $\square$

Proof of Theorem 1

The series at the right-hand side of (8) converges in mean square for every $\mathbf {x} \in \mathbb {M}^d$ since

$$\begin{aligned}&\mathsf {E}\left[ \left( \sum _{i=n_1}^{n_1+n_2} \mathsf {B}_i^{\frac{1}{2}} \mathbf {V}_i P_i^{ (\alpha , \beta ) } ( \cos \rho (\mathbf {x}, \mathbf {U} )) \right) \left( \sum _{j=n_1}^{n_1+n_2} \mathsf {B}_j^{\frac{1}{2}} \mathbf {V}_j P_j^{ (\alpha , \beta ) } ( \cos \rho (\mathbf {x}, \mathbf {U} )) \right) ^{\top }\right] \\&\quad = \sum _{i=n_1}^{n_1+n_2} \sum _{j=n_1}^{n_1+n_2} \mathsf {B}_i^{\frac{1}{2}} \mathsf {B}_j^{\frac{1}{2}} \mathsf {E}[ ( \mathbf {V}_i \mathbf {V}^{\top }_j)] \mathsf {E} \left[ \left( P_i^{ (\alpha , \beta ) } ( \cos \rho (\mathbf {x}, \mathbf {U} )) P_j^{ (\alpha , \beta ) } ( \cos \rho (\mathbf {x}, \mathbf {U} )) \right) \right] \\&\quad = \sum _{i=n_1}^{n_1+n_2} \mathsf {B}_i \sigma _i^2 \mathsf {E} \left[ \left( P_i^{ (\alpha , \beta ) } ( \cos \rho (\mathbf {x}, \mathbf {U} )) P_i^{ (\alpha , \beta ) } ( \rho (\mathbf {x}, \mathbf {U} )) \right) \right] \\&\quad = \sum _{i=n_1}^{n_1+n_2} \mathsf {B}_i P_i^{ (\alpha , \beta ) } ( 1) \\&\quad \rightarrow \mathbf {0}, ~~~~ \text{ as } ~ n_1, n_2 \rightarrow \infty , \end{aligned}$$

where the second equality follows from the independent assumption between $\{\, \mathbf {V}_n:n \in \mathbb {N}_0\, \}$ and $\mathbf {U}$, and the third from Lemma 3. Thus, (8) is an m-variate second-order random field. Its mean function is clearly identical to $\mathbf {0}$, and it covariance function is

$$\begin{aligned}&{{\,\mathrm{cov}\,}}\left( \sum _{i=0}^\infty \mathsf {B}_i^{\frac{1}{2}} \mathbf {V}_i P_i^{ (\alpha , \beta ) } ( \cos \rho (\mathbf {x}_1, \mathbf {U} )), ~ \sum _{j=0}^\infty \mathsf {B}_j^{\frac{1}{2}} \mathbf {V}_j P_j^{ (\alpha , \beta ) } ( \cos \rho (\mathbf {x}_2, \mathbf {U} )) \right) \\&\quad = \sum _{i=0}^\infty \sum _{j=0}^\infty \mathsf {B}_i^{\frac{1}{2}} \mathsf {B}_j^{\frac{1}{2}} \mathsf {E}\left[ ( \mathbf {V}_i \mathbf {V}^{\top }_j)] \mathsf {E}[ \left( P_i^{ (\alpha , \beta )} ( \cos \rho (\mathbf {x}, \mathbf {U} )) P_j^{ (\alpha , \beta ) } ( \cos \rho (\mathbf {x}, \mathbf {U} )) \right) \right] \\&\quad = \sum _{i=0}^\infty \mathsf {B}_i \sigma _i^2 \mathsf {E} \left[ \left( P_i^{ (\alpha , \beta ) } ( \cos \rho (\mathbf {x}_1, \mathbf {U} )) P_i^{ (\alpha , \beta ) } ( \cos \rho (\mathbf {x}_2, \mathbf {U} )) \right) \right] \\&\quad = \sum _{i=0}^\infty \mathsf {B}_i P_i^{ (\alpha , \beta ) } ( \cos \rho (\mathbf {x}_1, \mathbf {x}_2)), ~~~~ \mathbf {x}_1, \mathbf {x}_2 \in \mathbb {M}^d. \end{aligned}$$

Two distinct terms of (8) are obviously uncorrelated each other. $\square$

Proof of Theorem 2

It suffices to verify (10) to be a general form, since in Theorem 1 we already construct an m-variate isotropic random field on $\mathbb {M}^d$ whose covariance matrix function is (10). To this end, suppose that $\{\, \mathbf {Z}(\mathbf {x}):\mathbf {x} \in \mathbb {M}^d\, \}$ is an m-variate isotropic and mean square continuous random field. Then, for an arbitrary $\mathbf {a} \in \mathbb {R}^m$, $\{\, \mathbf {a}^{\top } \mathbf {Z}(\mathbf {x}):\mathbf {x} \in \mathbb {M}^d\, \}$ is a scalar isotropic and mean square continuous random field, so that its covariance function has to be of the form (1),

$$\begin{aligned} {{\,\mathrm{cov}\,}}\left( \mathbf {a}^{\top } \mathbf {Z}(\mathbf {x}_1), \mathbf {a}^{\top } \mathbf {Z}(\mathbf {x}_2) \right) = \sum _{n=0}^\infty b_n (\mathbf {a}) P_n^{ (\alpha , \beta ) } ( \cos \rho (\mathbf {x}_1, \mathbf {x}_2)), ~~~~ \mathbf {x}_1, \mathbf {x}_2 \in \mathbb {M}^d, \end{aligned}$$

(16)

where $\{\, b_n (\mathbf {a}):n \in \mathbb {N}_0\, \}$ is a sequence of nonnegative constants and $\sum \nolimits _{n=0}^\infty b_n (\mathbf {a}) P_n^{ (\alpha , \beta ) } ( 1 )$ converges. Similarly, for $\mathbf {b} \in \mathbb {R}^m$, we obtain

$$\begin{aligned}&\frac{1}{4} {{\,\mathrm{cov}\,}}( (\mathbf {a} +\mathbf {b})^{\top } \mathbf {Z}(\mathbf {x}_1), (\mathbf {a}+\mathbf {b})^{\top } \mathbf {Z}(\mathbf {x}_2) )\\&\quad = \sum _{n=0}^\infty b_n (\mathbf {a}+\mathbf {b}) P_n^{ (\alpha , \beta ) } ( \cos \rho (\mathbf {x}_1, \mathbf {x}_2)), \\&\frac{1}{4} {{\,\mathrm{cov}\,}}( (\mathbf {a} -\mathbf {b})^{\top } \mathbf {Z}(\mathbf {x}_1), (\mathbf {a}-\mathbf {b})^{\top } \mathbf {Z}(\mathbf {x}_2) )\\&\quad = \sum _{n=0}^\infty b_n (\mathbf {a}-\mathbf {b}) P_n^{ (\alpha , \beta ) } ( \cos \rho (\mathbf {x}_1, \mathbf {x}_2)), ~~ \mathbf {x}_1, \mathbf {x}_2 \in \mathbb {M}^d. \end{aligned}$$

Taking the difference between the last two equations yields

$$\begin{aligned}&\frac{1}{2} \left( \mathbf {a}^{\top } {{\,\mathrm{cov}\,}}( \mathbf {Z}(\mathbf {x}_1), \mathbf {Z}(\mathbf {x}_2) ) \mathbf {b}+ \mathbf {b}^{\top } {{\,\mathrm{cov}\,}}( \mathbf {Z}(\mathbf {x}_1), \mathbf {Z}(\mathbf {x}_2) )\mathbf {a} \right) \\&\quad =\frac{1}{2} \left( {{\,\mathrm{cov}\,}}( \mathbf {a}^{\top } \mathbf {Z}(\mathbf {x}_1), \mathbf {b}^{\top } \mathbf {Z}(\mathbf {x}_2) ) +{{\,\mathrm{cov}\,}}( \mathbf {b}^{\top } \mathbf {Z}(\mathbf {x}_1), \mathbf {a}^{\top } \mathbf {Z}(\mathbf {x}_2) )\right) \\&\quad =\sum \limits _{n=0}^\infty \left( b_n (\mathbf {a}+\mathbf {b}) -b_n (\mathbf {a}-\mathbf {b})\right) P_n^{ (\alpha , \beta ) } ( \cos \rho (\mathbf {x}_1, \mathbf {x}_2)), ~~ \mathbf {x}_1, \mathbf {x}_2 \in \mathbb {M}^d, \end{aligned}$$

or

$$\begin{aligned} \mathbf {a}^{\top } {{\,\mathrm{cov}\,}}( \mathbf {Z}(\mathbf {x}_1), \mathbf {Z}(\mathbf {x}_2) ) \mathbf {b} = \sum \limits _{n=0}^\infty (b_n (\mathbf {a}+\mathbf {b}) -b_n (\mathbf {a}-\mathbf {b})) P_n^{ (\alpha , \beta ) } ( \cos \rho (\mathbf {x}_1, \mathbf {x}_2)), ~~ \mathbf {x}_1, \mathbf {x}_2 \in \mathbb {M}^d, \end{aligned}$$

(17)

noticing that ${{\,\mathrm{cov}\,}}( \mathbf {Z}(\mathbf {x}_1), \mathbf {Z}(\mathbf {x}_2) )$ is a symmetric matrix. The form (10) of ${{\,\mathrm{cov}\,}}( \mathbf {Z}(\mathbf {x}_1), \mathbf {Z}(\mathbf {x}_2) )$ is now confirmed by letting the ith entry of $\mathbf {a}$ and the jth entry of $\mathbf {b}$ be 1 and the rest vanish in (17). It remains to verify the nonnegative definiteness of each $\mathsf {B}_n$ in (10). To do so, we multiply its both sides by $\mathbf {a}^{\top }$ from the left and $\mathbf {a}$ from the right, and obtain

$$\begin{aligned} \mathbf {a}^{\top } \mathsf {C} ( \mathbf {x}_1, \mathbf {x}_2 ) \mathbf {a} = \sum _{n=0}^\infty \mathbf {a}^{\top } \mathsf {B}_n \mathbf {a} P_n^{ (\alpha , \beta ) } \left( \cos \rho (\mathbf {x}_1, \mathbf {x}_2) \right) , ~~ \mathbf {x}_1, \mathbf {x}_2 \in \mathbb {M}^d, \end{aligned}$$

comparing which with (16) results in that $\mathbf {a}^{\top } \mathsf {B}_n \mathbf {a} \ge 0$ or the nonnegative definiteness of $\mathsf {B}_n$, $n \in \mathbb {N}_0$, and the convergence of $\sum \nolimits _{n=0}^\infty \mathbf {a}^{\top } \mathsf {B}_n \mathbf {a} P_n^{ (\alpha , \beta )} (1)$ or that of each entry of the matrix $\sum \nolimits _{n=0}^\infty \mathsf {B}_n P_n^{ (\alpha , \beta ) } (1)$. $\square$

Proof of Theorem 3

For a fixed $t \in \mathbb {T}$, consider a random field $\left\{ \, \mathbf {Z} (\mathbf {x}; 0) + \mathbf {Z} (\mathbf {x}; t):\right. \left. \mathbf {x} \in \mathbb {M}^d \,\right\}$. It is isotropic and mean square continuous on $\mathbb {M}^d$, with covariance matrix function

$$\begin{aligned}&{{\,\mathrm{cov}\,}}\left( \mathbf {Z} (\mathbf {x}_1; 0) + \mathbf {Z} (\mathbf {x}_1; t), ~ \mathbf {Z} (\mathbf {x}_2; 0) + \mathbf {Z} (\mathbf {x}_2; t) \right) \\&\quad =2\mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); 0) + \mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); t) + \mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); -t)\\&\quad =\sum _{n=0}^\infty \mathsf {B}_{n+} (t) P_n^{(\alpha , \beta ) } (\cos \rho (\mathbf {x}_1, \mathbf {x}_2)), ~~ \mathbf {x}_1, \mathbf {x}_2 \in \mathbb {M}^d, \end{aligned}$$

where the last equality follows from Theorem 2, $\{\, \mathsf {B}_{n+} (t):n \in \mathbb {N}_0\, \}$ is a sequence of nonnegative definite matrices, and $\sum \nolimits _{n=0}^\infty \mathsf {B}_{n+} (t) P_n^{(\alpha , \beta )} (1)$ converges. Similarly, we have

$$\begin{aligned}&{{\,\mathrm{cov}\,}}\left( \mathbf {Z} (\mathbf {x}_1; 0) - \mathbf {Z} (\mathbf {x}_1; t), ~ \mathbf {Z} (\mathbf {x}_2; 0) - \mathbf {Z} (\mathbf {x}_2; t) \right) \\&\quad =2 \mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); 0) - \mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); t) - \mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); -t)\\&\quad =\sum _{n=0}^\infty \mathsf {B}_{n-} (t) P_n^{ (\alpha , \beta ) } (\cos \rho (\mathbf {x}_1, \mathbf {x}_2)), \end{aligned}$$

and thus,

$$\begin{aligned}&\frac{\mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); t) + \mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); -t)}{2}\\&\quad =\frac{1}{4} [ 2 \mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); 0) + \mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); t) + \mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); -t)]\\&\qquad - \frac{1}{4} [ 2 \mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); 0) - \mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); t) - \mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); -t) ] \\&\quad =\sum _{n=0}^\infty \mathsf {B}_n (t) P_n^{(\alpha , \beta )} (\cos \rho (\mathbf {x}_1, \mathbf {x}_2)), ~~ \mathbf {x}_1, \mathbf {x}_2 \in \mathbb {M}^d, \end{aligned}$$

which confirms the format (11) for $\frac{\mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); t) + \mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); -t)}{2}$, with $B_n (t) =\frac{ \mathsf {B}_{n+} (t) - \mathsf {B}_{n-} (t)}{4}$, $n \in \mathbb {N}_0$. Obviously, $\mathsf {B}_n(t)$ is symmetric, and $\sum \nolimits _{n=0}^\infty \mathsf {B}_n (t) P_n^{ (\alpha , \beta ) } (1)$ converges. Moreover, (11) is the covariance matrix function of an m-variate isotropic random field $\left\{ \, \frac{ \mathbf {Z} (\mathbf {x}; t)+\tilde{\mathbf {Z}} (\mathbf {x}; -t)}{\sqrt{2}}:\mathbf {x} \in \mathbb {M}^d, t \in \mathbb {T}\, \right\}$, where $\{ \,\tilde{\mathbf {Z}} (\mathbf {x}; t):\mathbf {x} \in \mathbb {M}^d, t \in \mathbb {T} \,\}$ is an independent copy of $\{\, \mathbf {Z} (\mathbf {x}; t):\mathbf {x} \in \mathbb {M}^d, t \in \mathbb {T}\, \}$. In fact,

$$\begin{aligned}&{{\,\mathrm{cov}\,}}\left( \frac{ \mathbf {Z} (\mathbf {x}_1; t_1)+\tilde{\mathbf {Z}} (\mathbf {x}_1; -t_1)}{\sqrt{2}}, ~ \frac{ \mathbf {Z} (\mathbf {x}_2; t_2)+\tilde{\mathbf {Z}} (\mathbf {x}_2; -t_2)}{\sqrt{2}} \right) \\&\quad =\frac{\mathsf {C} (\rho (\mathbf {x}_1, \mathbf {x}_2); t_1-t_2) +\mathsf {C} ( \rho (\mathbf {x}_1, \mathbf {x}_2); t_2-t_1)}{2} \\&\quad =\sum _{k=0}^\infty \mathsf {B}_{k} (t_1-t_2) P_k^{ (\alpha , \beta )} (\cos \rho (\mathbf {x}_1, \mathbf {x}_2)) \end{aligned}$$

with $\mathbf {x}_1$, $\mathbf {x}_2 \in \mathbb {M}^d$, $t_1$, $t_2 \in \mathbb {T}$.

For each fixed $n \in \mathbb {N}_0$, in order to verify that $\mathsf {B}_n (t)$ is a stationary covariance matrix function on $\mathbb {T}$, we consider an m-variate stochastic process

$$\begin{aligned} \mathbf {W}_n (t) = \int _{\mathbb {M}^d} \frac{ \mathbf {Z} (\mathbf {x}; t)+\tilde{\mathbf {Z}} (\mathbf {x}; -t)}{\sqrt{2}} P_n^{ (\alpha , \beta ) } ( \cos \rho (\mathbf {x}, \mathbf {U}) ) \mathrm{d} \mathbf {x}, \quad t \in \mathbb {T}, \end{aligned}$$

where $\{\, \tilde{\mathbf {Z}} (\mathbf {x}; t):\mathbf {x} \in \mathbb {M}^d, t \in \mathbb {T}\, \}$ is an independent copy of $\{\, \mathbf {Z} (\mathbf {x}; t):\mathbf {x} \in \mathbb {M}^d, t \in \mathbb {T}\, \}$, $\mathbf {U}$ is a random vector uniformly distributed on $\mathbb {M}^d$, and $\mathbf {U}$, $\{\, \mathbf {Z} (\mathbf {x}; t):\mathbf {x} \in \mathbb {M}^d, t \in \mathbb {T}\, \}$ and $\{\, \tilde{\mathbf {Z}} (\mathbf {x}; t):\mathbf {x} \in \mathbb {S}^d, t \in \mathbb {T}\, \}$ are independent. By Lemma 2, the mean function of $\{\, \mathbf {W}_n (t):t \in \mathbb {T}\, \}$ is

$$\begin{aligned} \mathsf {E} [ \mathbf {W}_n (t)] = \left\{ \begin{array}{ll} \sqrt{2} P^{(\alpha ,\beta )}_0(1)\omega _d \mathsf {E} [\mathbf {Z} (\mathbf {x}; t)], ~ &{} ~ n= 0, \\ 0, ~ &{} ~ n \in \mathbb {N}, \end{array} \right. \end{aligned}$$

and its covariance matrix function is by Lemmas 2 and 3

$$\begin{aligned}&{{\,\mathrm{cov}\,}}( \mathbf {W}_n(t_1), ~ \mathbf {W}_n (t_2) ) \\&\quad = \frac{1}{\omega _d} {{\,\mathrm{cov}\,}}\left( \int _{\mathbb {M}^d} \frac{ \mathbf {Z} (\mathbf {x}; t_1)+\tilde{\mathbf {Z}} (\mathbf {x}; -t_1)}{\sqrt{2}} P_n^{ (\alpha , \beta ) } ( \cos \rho ( \mathbf {x}, \mathbf {U}) ) \mathrm{d} \mathbf {x},\right. \\&\qquad \left. \int _{\mathbb {M}^d} \frac{ \mathbf {Z} (\mathbf {y}; t_2)+\tilde{\mathbf {Z}} (\mathbf {y}; -t_2)}{\sqrt{2}} P_n^{ (\alpha , \beta ) } ( \cos \rho ( \mathbf {y}, \mathbf {U})) \mathrm{d} \mathbf {y} \right) \\&\quad =\frac{1}{\omega _d} \int _{\mathbb {M}^d} {{\,\mathrm{cov}\,}}\left( \int _{\mathbb {M}^d} \frac{ \mathbf {Z} (\mathbf {x}; t_1)+\tilde{\mathbf {Z}} (\mathbf {x}; -t_1)}{\sqrt{2}} P_n^{ (\alpha , \beta ) } ( \cos \rho ( \mathbf {x}, \mathbf {U}) )\mathrm{d} \mathbf {x}, ~ \right. \\&\qquad \left. ~~~~~~~~~~ \int _{\mathbb {M}^d} \frac{ \mathbf {Z} (\mathbf {y}; t_2)+\tilde{\mathbf {Z}} (\mathbf {y}; -t_2)}{\sqrt{2}} P_n^{ (\alpha , \beta ) } (\cos \rho ( \mathbf {y}, \mathbf {u}) ) \mathrm{d} \mathbf {y} \right) d \mathbf {u} \\&\quad =\frac{1}{\omega _d} \int _{\mathbb {M}^d} \int _{\mathbb {M}^d} \int _{\mathbb {M}^d} {{\,\mathrm{cov}\,}}\left( \frac{ \mathbf {Z} (\mathbf {x}; t_1)+\tilde{\mathbf {Z}} (\mathbf {x}; -t_1)}{\sqrt{2}}, ~ \frac{ \mathbf {Z} (\mathbf {y}; t_2)+\tilde{\mathbf {Z}} (\mathbf {y}; -t_2)}{\sqrt{2}} \right) \\&\qquad \times P_n^{ (\alpha , \beta ) } (\cos \rho ( \mathbf {x}, \mathbf {u}) ) P_n^{ (\alpha , \beta ) } (\cos \rho ( \mathbf {y}, \mathbf {u}) ) \mathrm{d} \mathbf {x} \mathrm{d} \mathbf {y} \mathrm{d} \mathbf {u} \\&\quad =\int _{\mathbb {M}^d} \int _{\mathbb {M}^d} \int _{\mathbb {M}^d} \frac{ \mathsf {C} (\rho (\mathbf {x}, \mathbf {y}); t_1-t_2)+\mathsf {C} ( \rho (\mathbf {x}, \mathbf {y}); t_2-t_1)}{2\omega _d}\\&\qquad \times P_n^{ (\alpha , \beta ) } ( \cos \rho ( \mathbf {x}, \mathbf {u}) ) P_n^{ (\alpha , \beta ) } ( \cos \rho ( \mathbf {y}, \mathbf {u}) ) \mathrm{d} \mathbf {x} \mathrm{d} \mathbf {y} \mathrm{d} \mathbf {u} \\&\quad =\frac{1}{\omega _d} \int _{\mathbb {M}^d} \int _{\mathbb {M}^d} \int _{\mathbb {M}^d} \sum _{k=0}^\infty \mathsf {B}_{k} (t_1-t_2) P_k^{ (\alpha , \beta ) } ( \cos \rho ( \mathbf {x}, \mathbf {y}))\\&\qquad \times P_n^{ (\alpha , \beta ) } (\cos \rho ( \mathbf {x}, \mathbf {u})) P_n^{ (\alpha , \beta ) } (\cos \rho ( \mathbf {y}, \mathbf {u})) \mathrm{d} \mathbf {x} \mathrm{d} \mathbf {y} \mathrm{d} \mathbf {u} \\&\quad =\frac{1}{\omega _d} \sum _{k=0}^\infty \mathsf {B}_{k} (t_1-t_2) \int _{\mathbb {M}^d} \int _{\mathbb {M}^d} \int _{\mathbb {M}^d} P_k^{ (\alpha , \beta ) } (\cos \rho ( \mathbf {x}, \mathbf {y}))\\&\qquad \times P_n^{ (\alpha , \beta ) } ( \cos \rho ( \mathbf {x}, \mathbf {u}) ) \mathrm{d} \mathbf {x} P_n^{ (\alpha , \beta ) } (\cos \rho ( \mathbf {y}, \mathbf {u}) ) \mathrm{d} \mathbf {y} \mathrm{d} \mathbf {u} \\&\quad = \frac{1}{\omega _d} \mathsf {B}_{n} (t_1-t_2) \int _{\mathbb {M}^d} \frac{1}{a_n^2} \int _{\mathbb {M}^d} P_n^{ (\alpha , \beta ) } (\cos \rho ( \mathbf {y}, \mathbf {u})) P_n^{ (\alpha , \beta ) } (\cos \rho ( \mathbf {y}, \mathbf {u})) \mathrm{d} \mathbf {y} \mathrm{d} \mathbf {u} \\&\quad = \frac{1}{\omega _d} \mathsf {B}_{n} (t_1-t_2) \int _{\mathbb {M}^d} \left( \frac{\omega _d}{a_n^2} \right) ^2 P_n^{(\alpha , \beta ) } (1) \mathrm{d} \mathbf {u} \\&\quad =\mathsf {B}_{n} (t_1-t_2) \left( \frac{\omega _d}{a_n^2} \right) ^2 P_n^{ (\alpha , \beta ) } (1), ~~~~~~ t_1, t_2 \in \mathbb {T}, \end{aligned}$$

which implies that $\mathsf {B}_n (t)$ is a stationary covariance matrix function on $\mathbb {T}$. $\square$

Proof of Theorem 4

The convergent assumption of $\sum \nolimits _{n=0}^\infty \mathsf {B}_n (0) P_n^{ (\alpha , \beta ) } (1)$ ensures the uniform and absolute convergence of the series at the right-hand side of (12). If $\{\, \mathsf {B}_n (t):n \in \mathbb {N}_0\, \}$ is a sequence of stationary covariance matrix function on $\mathbb {T}$, then each term of the series at the right-hand side of (12) is the product of a stationary covariance matrix function $\mathsf {B}_n (t)$ on $\mathbb {T}$ and an isotropic covariance function $P_n^{ (\alpha , \beta ) } (\cos \rho (\mathbf {x}_1, \mathbf {x}_2)$ on $\mathbb {M}^d$, and thus, (12) can be treated [21] as the covariance matrix function of an m-variate random field on $\mathbb {M}^d \times \mathbb {T}$.

On the other hand, assume that (12) is the covariance matrix function of an m-variate random field $\{\, \mathbf {Z} (\mathbf {x}; t):\mathbf {x} \in \mathbb {M}^d, t \in \mathbb {T}\, \}$. The convergence of $\sum \nolimits _{n=0}^\infty \mathsf {B}_n (0) P_n^{ (\alpha , \beta ) } (1)$ results from the existence of $\mathsf {C} (0; 0) = {{\,\mathrm{\mathsf {Var}}\,}}[Z (\mathbf {x}; t)]$. In order to show that $\mathsf {B}_n (t)$ is a stationary covariance matrix function on $\mathbb {T}$ for each fixed $n \in \mathbb {N}_0$, consider an m-variate stochastic process

$$\begin{aligned} \mathbf {W}_n (t) = \int _{\mathbb {M}^d} \mathbf {Z} (\mathbf {x}; t) P_n^{ (\alpha , \beta ) } ( \cos \rho (\mathbf {x}, \mathbf {U})) \mathrm{d} \mathbf {x},\quad t \in \mathbb {T}, \end{aligned}$$

where $\mathbf {U}$ is a random vector uniformly distributed on $\mathbb {M}^d$ and is independent with $\{\, \mathbf {Z} (\mathbf {x}; t):\mathbf {x} \in \mathbb {M}^d, t \in \mathbb {T} \,\}$. Similar to that in the proof of Theorem 3, applying Lemmas 2 and 3 we obtain that the covariance matrix function of $\{\, \mathbf {W}_n (t):t \in \mathbb {T}\, \}$ is positively propositional to $\mathsf {B}_n (t)$; more precisely,

$$\begin{aligned} {{\,\mathrm{cov}\,}}( \mathbf {W}_n (t_1), \mathbf {W}_n (t_2) ) = \mathsf {B}_{n} (t_1-t_2) \left( \frac{\omega _d}{a_n^2} \right) ^2 P_n^{ (\alpha , \beta ) } (1), ~~~~~~ t_1, t_2 \in \mathbb {T}, \end{aligned}$$

which implies that $\mathsf {B}_n (t)$ is a stationary covariance matrix function on $\mathbb {T}$. $\square$

Proof of Theorem 5

The convergent assumption of $\sum \nolimits _{n=0}^\infty \mathsf {B}_n (0) P_n^{ (\alpha , \beta ) } (1)$ ensures the mean square convergence of the series at the right-hand side of (13), since

$$\begin{aligned}&\mathsf {E}\left[ \left( \sum _{i=n_1}^{n_1+n_2} \mathbf {V}_i (t) P_i^{ (\alpha , \beta ) } ( \cos \rho ( \mathbf {x}, \mathbf {U}) ) \right) \left( \sum _{j=n_1}^{n_1+n_2} \mathbf {V}_j (t) P_j^{ (\alpha , \beta )} ( \cos \rho ( \mathbf {x}, \mathbf {U})) \right) ^{\top }\right] \\&\quad =\mathsf {E} \left[ \sum _{i=n_1}^{n_1+n_2} \sum _{j=n_1}^{n_1+n_2} \mathbf {V}_i (t) \mathbf {V}^{\top }_j (t) P_i^{ (\alpha , \beta ) } (\cos \rho ( \mathbf {x}, \mathbf {U})) P_j^{ (\alpha , \beta ) } (\cos \rho ( \mathbf {x}, \mathbf {U})) \right] \\&\quad = \sum _{i=n_1}^{n_1+n_2} \sum _{j=n_1}^{n_1+n_2} \mathsf {E} [\mathbf {V}_i (t) \mathbf {V}^{\top }_j (t) ] \mathsf {E} \left[ P_i^{ (\alpha , \beta ) } (\cos \rho ( \mathbf {x}, \mathbf {U})) P_j^{ (\alpha , \beta ) } (\cos \rho ( \mathbf {x}, \mathbf {U})) \right] \\&\quad = \omega _d \sum _{i=n_1}^{n_1+n_2} \mathsf {B}_i (0) P_i^{ (\alpha , \beta ) } (1) \\&\quad \rightarrow 0, ~~~~~~~ \text{ as } ~ n_1, n_2 \rightarrow \infty , \end{aligned}$$

where the second equality follows from the independent assumption between $\mathbf {U}$ and $\{\, \mathbf {V}_n (t):n \in \mathbb {N}_0\, \}$, and the third one from Lemma 3. Applying Lemma 3 we obtain the mean and covariance matrix functions of $\{\, \mathbf {Z} (\mathbf {x}; t):\mathbf {x} \in \mathbb {M}^d, t \in \mathbb {T}\, \}$, under the independent assumption among $\mathbf {U}$ and $\{\, \mathbf {V}_n (t):n \in \mathbb {N}_0\, \}$,

$$\begin{aligned} \mathsf {E} [\mathbf {Z} (\mathbf {x}; t )] = \sum _{n=0}^\infty \mathsf {E} [\mathbf {V}_n (t)] \mathsf {E} \left[ P_n^{ (\alpha , \beta ) } (\cos \rho ( \mathbf {x}, \mathbf {U}))\right] = \mathbf {0}, ~~~ \mathbf {x} \in \mathbb {M}^d, t \in \mathbb {T}, \end{aligned}$$

and

$$\begin{aligned}&{{\,\mathrm{cov}\,}}( \mathbf {Z} (\mathbf {x}_1; t_1), \mathbf {Z} ( \mathbf {x}_2; t_2) )\\&\quad = {{\,\mathrm{cov}\,}}\left( \sum _{i=0}^\infty \mathbf {V}_i (t_1) P_i^{ (\alpha , \beta ) } (\cos \rho ( \mathbf {x}_1, \mathbf {U})), ~ \sum _{j=0}^\infty \mathbf {V}_j (t_2) P_j^{ (\alpha , \beta ) } (\cos \rho ( \mathbf {x}_2, \mathbf {U})) \right) \\&\quad =\sum _{i=0}^\infty \sum _{j=0}^\infty \mathsf {E} [ \mathbf {V}_i (t_1) \mathbf {V}^{\top }_j (t_2) ] \mathsf {E} \left[ P_i^{ (\alpha , \beta ) } ( \cos \rho ( \mathbf {x}_1, \mathbf {U})) P_j^{ (\alpha , \beta ) } (\cos \rho ( \mathbf {x}_2, \mathbf {U})) \right] \\&\quad =\sum _{n=0}^\infty \mathsf {B}_n (t_1-t_2)\frac{1}{a^2_n} P_n^{ (\alpha , \beta ) } ( \cos \rho (\mathbf {x}_1, \mathbf {x}_2) ), ~~~~~~~~~~ \mathbf {x}_1, \mathbf {x}_2 \in \mathbb {M}^d, ~ t_1, t_2 \in \mathbb {T}. \end{aligned}$$

The latter is obviously isotropic and continuous on $\mathbb {M}^d$ and stationary on $\mathbb {T}$. $\square$