Multivariate random fields on some homogeneous spaces
The generalized continuous random fields of second order with values in arbitrary complex normed space X in the case when their arguments belong to homogeneous space with compact transformation group G are considered. Such fields are harmonizable in some sense. The spectral representations of homoge...
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irk-123456789-45722009-12-08T12:00:35Z Multivariate random fields on some homogeneous spaces Ponomarenko, O. Perun, Y. The generalized continuous random fields of second order with values in arbitrary complex normed space X in the case when their arguments belong to homogeneous space with compact transformation group G are considered. Such fields are harmonizable in some sense. The spectral representations of homogeneous random fields in X and G-invariant positive definite operator-valued kernels are obtained. The special case of random fields with values in complex Hilbert space and random fields on three-dimensional spheres are also studied. 2008 Article Multivariate random fields on some homogeneous spaces / O. Ponomarenko, Y. Perun // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 3-4. — С. 104-113. — Бібліогр.: 12 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4572 en Інститут математики НАН України |
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The generalized continuous random fields of second order with values in arbitrary complex normed space X in the case when their arguments belong to homogeneous space with compact transformation group G are considered. Such fields are harmonizable in some sense. The spectral representations of homogeneous random fields in X and G-invariant positive definite operator-valued kernels are obtained. The special case of random fields with values in complex Hilbert space and random fields on three-dimensional spheres are also studied. |
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Ponomarenko, O. Perun, Y. |
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Ponomarenko, O. Perun, Y. Multivariate random fields on some homogeneous spaces |
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Ponomarenko, O. Perun, Y. |
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Ponomarenko, O. |
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Multivariate random fields on some homogeneous spaces |
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Multivariate random fields on some homogeneous spaces |
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Multivariate random fields on some homogeneous spaces |
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Multivariate random fields on some homogeneous spaces |
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Multivariate random fields on some homogeneous spaces |
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multivariate random fields on some homogeneous spaces |
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Інститут математики НАН України |
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2008 |
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Multivariate random fields on some homogeneous spaces / O. Ponomarenko, Y. Perun // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 3-4. — С. 104-113. — Бібліогр.: 12 назв.— англ. |
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2025-07-02T07:46:58Z |
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Theory of Stochastic Processes
Vol. 14(30), no.3-4, 2008, pp.104-113
OLEKSANDER PONOMARENKO AND YURIY PERUN
MULTIVARIATE RANDOM FIELDS ON SOME
HOMOGENEOUS SPACES
The generalized continuous random fields of second order with val-
ues in arbitrary complex normed space X in the case when their
arguments belong to homogeneous space with compact transforma-
tion group G are considered. Such fields are harmonizable in some
sense. The spectral representations of homogeneous random fields
in X and G-invariant positive definite operator-valued kernels are
obtained. The special case of random fields with values in complex
Hilbert space and random fields on three-dimensional spheres are
also studied.
1. Introduction
The central role in the theory of second-order random functions (pro-
cesses and fields) plays different representations of these functions in the
form of stochastic integrals and series.
Such representations for complex-valued and multivariate finite-dimen-
sional homogeneous random fields over different homogeneous spaces were
studied by A.M.Yaglom [1]. The theory of integral representations of mul-
tivariate generalized random functions of second order with values in linear
topological spaces was studied in [2].
In this paper we consider the representations of generalized random fields
of second order with values in complex normed and Hilbert spaces defined
over compact groups and compact homogeneous spaces by mean of random
series. The main attention is devoted to representations of homogeneous
second order random fields and invariant operator-valued positive definite
functions.
Let X be a complex normed space and X∗ be its topological dual space
endowed by strong topology. Denote by L2(Ω) Hilbert space of all complex
2000 Mathematics Subject Classifications. Primary 60G10, 60G57.
Key words and phrases. Generalized random fields of second order in normed spaces,
homogeneous space with compact transformation group, unitary representations, spectral
representations, harmonizable field, invariant positive definite operator-valued kernels.
104
MULTIVARIATE RANDOM FIELDS 105
second order random variables defined on some probability space (Ω,F ,P)
(inner product in L2(Ω) is equal to covariance of r.v. from L2(Ω)) endowed
by strong topology.
The generalized random element Ξ on Ω in X is continuous linear opera-
tor Ξ : X∗ → L2(Ω). Every such element is generated by some usual random
element ξ on Ω with values in some extension of the space X (see [3]). The
space of generalized random elements in X will denoted by L(X∗, L2(Ω))
and endowed by strong operator topology.
Let L(X∗, X∗∗) be a space of all antilinear (or conjugate linear) contin-
uous operators from X∗ into the X∗∗ (second dual space for X) endowed
by weak operator topology. Define the expectation EΞ ∈ X∗∗ of random
element Ξ ∈ L(X∗, L2(Ω)) by the equality
(EΞ)(x∗) = E(Ξx∗), x∗ ∈ X∗ (1)
and covariance operator [Ξ,Φ] ∈ L(X∗, X∗∗) of elements Ξ, Φ ∈ L(X∗, L2(Ω))
by the equality
E(Ξx∗)(Φy∗) = ([Ξ,Φ]y∗)(x∗), x∗, y∗ ∈ X∗ (2)
Note that operator [Ξ,Ξ] is nonnegative in L(X∗, X∗∗), i.e. for all x∗ ∈
X∗ : ([Ξ,Ξ]x∗)(x∗) ≥ 0.
Let T be a topological Hausdorff space. The generalized random field of
second order Ξt t ∈ T inX is mapping of T into L(X∗, L2(Ω)). In further we
assume that this mapping is continuous. Denote by R(t, s) the covariance
function of random field Ξt, t ∈ T ,
R(t, s) = [Ξt,Ξs], t, s ∈ T (3)
Then R(t, s) is continuous L(X∗, X∗∗)-valued kernel on T .
Note that the class of covariance functions of the set of all second order
random fields in X coinsides with the set of all continuous positive definite
L(X∗, X∗∗)-valued operator kernels on T , i.e. for all n ∈ N, elements x∗i ∈
X∗, ti ∈ T, i = 1, ..., n
n∑
i=1
n∑
j=1
(R(ti, tj)x
∗
j )(x
∗
i ) ≥ 0 (4)
(see [4]). This fact give to us the opportunity to obtain spectral represen-
tations of some classes of invariant operator positive definite functions on
compact groups and compact homogeneous spaces by mean of probabilistic
methods.
106 OLEKSANDER PONOMARENKO AND YURIY PERUN
2. Random fields on compact groups
The simplest type of compact homogeneous space is the group space
G = {g} consisting of the elements g of some compact group G. There are
two different families of continuous transformations of G namely left and
right shifts
V l
g : h→ gh and V r
g : h→ hg. (5)
Let Ξg, g ∈ G be a second order continuous random field in X over G.
The field Ξg will be called left (right) homogeneous if its mean function
EΞg and covariance function R(g, h) = [Ξg,Ξh], g, h ∈ G are invariant with
respect to all left shifts V l
g (right shifts V r
g ). Due to transitivity of transfor-
mation group V l
g (V
r
g ), g ∈ G we always have that EΞg = m = const and
that for all s ∈ G for left homogeneous field
[Ξg,Ξh] = [Ξsg,Ξsh] = R(h−1g), g, h ∈ G (6)
and for right homogeneous field
[Ξg,Ξh] = [Ξgs,Ξhs] = R(gh−1), g, h ∈ G. (7)
For two-way homogeneous field conditions (6) and (7) must be satisfied
simultaneously, so that the covariance function R such field must be a con-
stant for a class of conjugate elements of G:
R(h) = R(ghg−1), g, h ∈ G. (8)
In the further we assume that for homogeneous field Ξg the mean EΞg is
zero.
According to the theory of unitary representations of compact group G
(see[5],[6]) there exits the system Ug of not more than a countable nonequiv-
alent finite dimensional unitary irreducible representations Ug = {U (λ)
g , λ =
1, ...}, where U
(λ)
g are homomorphisms of the group G into group of unitary
matrices of finite order dλ, namely
U (λ)
g = ‖u(λ)
ij (g)‖dλ
i,j=1, dλ <∞, λ = 1, 2, ...
U
(λ)
gh = U (λ)
g U
(λ)
h , U
(λ)
g−1 = [U (λ)
g ]−1 = [U (λ)
g ]∗. (9)
The matrix elements u
(λ)
ij (g) of these representations satisfy the following
orthogonality relations∫
G
u
(λ)
ij (g)u
(μ)
kl (g)dg = δλμδikδjld
−1
λ , (10)
where δλμ is a Kroneker symbol and dg is unique normed invariant measure
on G,
∫
G
dg = 1.
MULTIVARIATE RANDOM FIELDS 107
Theorem 2.1 Let Ξg, g ∈ G be a generalized continuous random field of
second order in normed space X on compact group G. Then the field Ξg is
harmonizable with respect to the system Ug = {U (λ)
g } of unitary representa-
tions of G (9) in sense that Ξg admits expansion by the random series of
the form
Ξg =
∑
λ
dλ∑
i,j=1
u
(λ)
ij (g)Φ
(λ)
ji , (11)
where u
(λ)
ij (g) are the matrix elements of representations {U (λ)
g } and Φ
(λ)
ji ∈
L(X∗, L2(Ω)),
Φ
(λ)
ji = dλ
∫
G
u
(λ)
ij (g)Ξgdg, (12)
where integral in (12) is understood in sense of strong topology in
L(X∗, L2(Ω)).
The statement of this theorem is simple consequence of the fact that
the set of all matrix elements u
(λ)
ij (g) for 1 ≤ i, j ≤ dλ, λ = 1, 2, ... of
system {U (λ)
g } form a complete orthogonal system in the Hilbert space L2(G)
of complex functions over G whose square of modulus are integrable with
respect to dg.
Theorem 2.2 I. The continuous random field Ξg, g ∈ G of second order in
X is left homogeneous if and only if it admits the spectral representation of
the form (11) with random elements Φ
(λ)
ji which satisfy the conditions
[Φ
(λ)
ji ,Φ
(μ)
lk ] = δλμδikF
(λ)
jl , (13)
where
F
(λ)
jl =
∫
G
u
(λ)
jl (g)R(g)dg ∈ L(X∗, X∗∗), (14)
matrices F (λ) = ‖F (λ)
ij ‖dλ
i,j=1 are positive definite in the sense of similar to
(4), and such that the series∑
λ
∑
j
F
(λ)
jj =
∑
λ
Tr(F
(λ)) (15)
is convergent in L(X∗, X∗∗).
II.The covariance function R(g) of Ξg can be represented in the form
R(g) =
∑
λ
∑
j,l
u
(λ)
lj (g)F
(λ)
jl . (16)
Conversely, any L(X∗, X∗∗)-valued function of the form (16), where
F (λ) = ‖F (λ)
jl ‖ are positive definite operator matrices satisfying the condi-
tion of convergence (15) is a covariance function of some left homogeneous
108 OLEKSANDER PONOMARENKO AND YURIY PERUN
random field over G in X.
III. The foregoing situation with spectral representation (11) is quite anal-
ogous for right homogeneous random field Ξg in X, the only change being
that condition (13) is replaced by condition
[Φ
(λ)
ji ,Φ
(μ)
lk ] = δλμδjlF
(λ)
ik . (17)
Proof. The necessity of condition (13) in I. From the equalities (6), (12),
(10) and invariance of measure dg it easily follows that
[Φ
(λ)
ji ,Φ
(μ)
lk ] = δλμδikdλ
∫
G
ulj(g)R(g)dg = δλμδikF
(λ)
jl . (18)
Substituting (11) into (6) we can verify equality (16). The operator matrices
F (λ) are obviously all positive definite. Because u
(λ)
ik (e) = δik, where e is unit
of G, it follows that convergence of series (15) is condition for convergence
of right side of (11) and (16).
The sufficiency of condition (12) and II. If F (λ), λ = 1, 2, ... are arbitrary
positive definite (dλ × dλ) operator L(X∗, X∗∗) - valued matrices, then one
can always select Φ
(λ)
jl ∈ L(X∗, L2(Ω)) such that (13) satisfied. Under con-
dition of convergence of (15) the series in right side of (11) converges and
defines a random field Ξg in X for which [Ξhg,Ξh] is given by the formula
(16), i.e. Ξg is left homogeneous.
The proof of the part III of the theorem is analogous to the proof of
parts I and II.
Corollary 2.3.Continuous L(X∗, X∗∗) - valued operator function R(g)is
left positive definite, i.e. for all gi ∈ G, x∗i ∈ X∗, i = 1, ..., n, n ∈ N
n∑
i,j=1
[R(g−1
j gi)x
∗
j ](x
∗
i ) ≥ 0,
if and only if it admits representation of the form (16).
This result follows from characterization of the class of covariance func-
tions of random fields over G in X by condition (4) and theorem 2.2. This
is an operator version of Bochner theorem for complex positive definite
function over G [7].
Theorem 2.4.The continuous random field Ξg, g ∈ G in X is two-way
homogeneous if and only if its admits the spectral representation (11) with
random elements Φ
(λ)
jl ∈ L(X∗, L2(Ω)) satisfying the condition
[Φ
(λ)
ji ,Φ
(μ)
lk ] = δλμδjlδikF
(λ), (19)
where F (λ) is nonnegative operators in L(X∗, X∗∗) such that series
∑
λ
dλF
(λ)
is convergent in L(X∗, X∗∗).
MULTIVARIATE RANDOM FIELDS 109
The covariance function R(g) of two-way homogeneous random field
Ξg, g ∈ G in X is represented in the form
R(g) =
∑
λ
χ(λ)(g)F (λ), (20)
where χ(λ) = Tr(U
(λ)
g ) are characters of the group G.
This theorem is consequence of part I, II and III of theorem 2.2.
Corollary 2.5.The expansion (20) is general characterization of L(X∗,X∗∗)
valued operator continuous positive definite functions over G which is in-
variant with respect to two-way shifts. This expansion is operator version
of corresponding result of Bochner for complex functions [7].
3. Random fields on homogeneous spaces
Let Q = {q} be a homogeneous space with transitive transformation
group G = {g}. It is well known that Q can be identified with the set of left
cosets G/K, where K is a stationary subgroup of G, which leaves invariant
some point q0 ∈ Q, and for q = hK, h ∈ G the actoin gq = ghK. The
topology of G induced naturally a topology in Q [6]. Functions in Q are
continuous if and only if the corresponding functions assuming a constant
value over elements from G/K are continuous on G.
Second order random field Ξq, q ∈ Q in X is a continuous mapping of Q
into L(X∗, L2(Ω)). The field Ξq is called homogeneous if for all g ∈ G, p, q ∈
Q
EΞgq = EΞq = const, [Ξgq,Ξgp] = [Ξq,Ξp] = R(q, p). (21)
In the following we suppose that EΞq = 0.
It is obvious that class of homogeneous random fields on Q coincides
with the class of homogeneous random fields on G which are constant over
all left cosets modulo K.
Now assume that the group G is compact. In order to obtain spectral
representation for Ξq we must use general theory of spherical functions (or
harmonics)on Q (see [8], [9]). Let us consider the complete system U
(λ)
g
of unitary continuous nonequivalent representation of G (9) and choose
in the space of irreducible representations a basis such that to obtain the
irreducible representation of K. In order that a matrix element u
(λ)
ij (g) be
a constant on all elements of G/K we must have
u
(λ)
ij (gk) =
∑
m
u
(λ)
im (g)u
(λ)
mj(k) = u
(λ)
ij (g), g ∈ G, k ∈ K. (22)
It means that the equalities
u
(λ)
mj(k) = δmj , m = 1, ..., dλ, k ∈ K
110 OLEKSANDER PONOMARENKO AND YURIY PERUN
take place. So elements u
(λ)
ij (g)which are constant on cosets of G/K fill
out the column of U
(λ)
g corresponding to the identity representation of K.
If representation U
(λ)
g contains rλ times the identity representation of K,
suppose that in basis e1, ...edλ
these identity representations correspond to
the first rλ basis vectors, U
(λ)
k ej = ej , k ∈ K, j = 1, ..., rλ. In this case the
functions of q
ψ
(λ)
ij (q) = u
(λ)
ij (g), i = 1, ..., dλ, j = 1, ..., rλ, λ = 1, 2, ... (23)
are called spherical functions on Q while the functions
ψ
(λ)
ij (q) = u
(λ)
ij (g), i = 1, ..., rλ, j = 1, ..., rλ, λ = 1, 2, ... (24)
are called zonal spherical functions. The functions (24)are constant on all
spheres with center at q0, i.e. sets of points kq, k ∈ K, q ∈ Q (set kq, k ∈ K
is a sphere with center q0(= K) and passing through the point q). The
zonal function ψ
(λ)
ij (q) depends only on the invariants of the ordered pair of
points q and q0 which remain unaltered under all transformations g ∈ G,
i.e. on the composite distance from q to q0:
ψ
(λ)
ij (q) = ψ
(λ)
ij (q, q0) = ψ
(λ)
ij (gq, gq0), g ∈ G, i, j = 1, ..., rλ, λ = 1, 2, ... (25)
Theorem 3.1. The continuous random field Ξq, q ∈ Q of second order in
X on compact homogeneous space Q = G/K is harmonizable in the sense
that it admits the representation
Ξq =
∑
λ
dλ∑
i=1
rλ∑
j=1
ψ
(λ)
ij (q)Φ
(λ)
ji , (26)
where ψ
(λ)
ij are spherical harmonics (23)on Q and Φ
(λ)
ji ∈ L(X∗, L2(Ω)) have
the form
Φ
(λ)
ij = (
∫
Q
|ψ(λ)
ij (q)|2dq)−1
∫
Q
ψ
(λ)
ij (q)Ξqdq, (27)
where dq is G-invariant measure on Q.
The statement of the theorem is consequence of the fact that accord-
ing to general theory of spherical harmonics the functions (23) represents
a complete orthogonal system in the Hilbert space L2(Q) of complex func-
tions ϕ(q), q ∈ Q such that |ϕ(q)|2 is integrable with respect to G-invariant
measure dq on Q. So only the harmonics (23) enter into the expansion of
the function ϕ(g) which is constant over all cosets from G/K.
Theorem 3.2 The continuous homogeneous random field Ξq, q ∈ Q on
compact homogeneous space Q = G/K in X admits the spectral represen-
tation (26) if and only if random elements Φ
(λ)
ji ∈ L(X∗, L2(Ω)) satisfy the
relations
[Φ
(λ)
ji ,Φ
(μ)
lk ] = δλμδikF
(λ)
jl , (28)
MULTIVARIATE RANDOM FIELDS 111
where operators F
(λ)
jl ∈ L(X∗, X∗∗)
The covariance function R(q, p) = [Ξq,Ξp] of such field Ξqcan be repre-
sented in the form
R(q, p) =
∑
λ
rλ∑
j,l=1
ψ
(λ)
lj (q, p)F
(λ)
jl (29)
where ψ
(λ)
lj (q, p) are the function (25). Conversely, any L(X∗, X∗∗)-valued
function of the form (29), where ‖F (λ)
jl ‖ are positive definite matrices in
L(X∗, X∗∗) such that series (29) converges is a covariance function of some
homogeneous field Ξq, q ∈ Q.
This theorem is consequence of foregoing theory of spherical harmonics
in view of theorems 2.2 and 3.1.
Corollary 3.3. The representation (29) gives general form of all continuous
G-invariant positive definite operator L(X∗, X∗∗)-valued kernels on Q =
G/K. This is operator version of related result of Bochner [7] for complex-
valued functions.
4. Some special cases
The special case of generalized random field in complex normed space
X is such field in complex Hilbert space H with inner product (·|·) and
strong topology. In this case it is more appropriate to give the definition
of generalized random element of second order Ξ in H as continuous linear
mapping from H into L2(Ω),Ξ ∈ L(H,L2(Ω)), the definition of expectation
EΞ as vector of H , for which
E(Ξx) = (x|EΞ), x ∈ H, (30)
and the definition of covariance operator [Ξ,Φ] of elements Ξ, Φ ∈
L(H,L2(Ω)) as linear continuous mapping of H into H which satisfy the
relation
([Ξ,Φ]x|y) = (E(Ξx)(Φy), x, y ∈ H. (31)
So covariance [Ξ,Φ] belongs to the space of bounded linear operator
L(H,H) = L(H) in space H . Every generalized random elements Ξ ∈
L(H,L2(Ω)) is generated by some usual random element in some quasinu-
clear extension of H (see [3]).
In the further we assume that the space L(H,L2(Ω)) is endowed by
strong operator topology and the space L(H) is endowed by weak operator
topology.
The second order generalized random field Ξt, t ∈ T over some topolog-
ical space T is a continuous mapping of T into L(H,L2(Ω)). The class of
covariance functions of such fields, R(t, s) = [Ξt,Ξs], t, s ∈ T coinsides with
112 OLEKSANDER PONOMARENKO AND YURIY PERUN
the class of continuous L(H)-valued operator positive definite kernels, i.e.
for all ti ∈ T, xi ∈ H, i = 1, ..., n, n ∈ N
n∑
i,j=1
(R(ti, tj)xi|xj) ≥ 0. (32)
The all results section 2 and 3 of this paper are valid for corresponding
generalized random fields over G and G/K in Hilbert space H and positive
definite invariant L(H)-valued operator kernels with using instead of space
L(X∗, L2(Ω)) of generalized random element in X the space L(H,L2(Ω))
of such elements in H and instead of covariance operators from L(X∗, X∗∗)
the covariance operators from L(H).
In the case, when Q is sphere S2 in three-dimensional Euclidean space R3
with transformation group G as the group of all rotations g around center
of sphere 0, the homogeneous random field Ξθ,ϕ, (θ, ϕ) ∈ S2 in X admits the
expansion in the form of series
Ξθ,ϕ =
∞∑
l=0
l∑
m=−l
Y m
l (θ, ϕ)Φl
m (33)
in accordance of theorem 3.2, where Y m
l (θ, ϕ) are spherical harmonics and
Φl
m are random elements from L(X∗, L2(Ω)) with covariances
[Φl
m,Φ
k
j ] = δmjδlkFm, (34)
where Fm are nonnegative operators from L(X∗, X∗∗).
It follows from the equalities (33) and (34) and the addition theorem
of associated Legendre functions that corresponding covariance function R
of the field Ξθ,ϕ depends only on angular distance θ1,2 between the points
(θ1, ϕ1), (θ2, ϕ2) ∈ S2, i.e.
R(θ1,2) = [Ξ(θ1,ϕ1),Ξ(θ2,ϕ2)]
and has representation of the form
R(θ1,2) =
m∑
l=0
Pl(cosθ1,2)Bl, (35)
where
Pl(cosθ1,2) =
2
2l + 1
l∑
m=−l
Y m
l (θ1ϕ1)Y
m
l (θ2ϕ2), Bl =
2l + 1
2
Fl.
Conversely, for every Φm
l and Fl satisfying (34) and such that the series
(35) converges the field (33) is homogeneous and the function (35) is a
covariance function of a homogeneous random field over S2.
MULTIVARIATE RANDOM FIELDS 113
Note that representations (33) and (35)are multivariate analogue of re-
sults of Obukhov [10] (see also [11]) for complex-valued fields on S2. Ex-
pansion (35) gives general form of G-invariant positive definite L(X∗, X∗∗)-
valued kernel on S2. This is a operator version of result of Shoenberg [12]
for complex-valued case.
References
1. Yaglom A.M., Second-order homogeneous random fields. Proc. Fourth
Berkeley Symp. Math.Stat. and Probab., 2 (1961), p.593-622.
2. Ponomarenko O.I., Integral representation of random functions with values
in locally convex spaces. Theory Probab. and Math. Stat., N46, (1992),
p.132–141.
3. Ponomarenko O.I., Random linear functionals of Second Order I. Theory
Probab. and Math. Stat., N54, (1997), p.145–154.
4. Ponomarenko O.I., Infinite-dimensional random fields on semigroups. The-
ory Probab. and Math. Stat., 30, (1985), p.153–158.
5. Pontrjagin L., Topological groups. Princeton Univ. Press,(1939).
6. Weil A., L‘integration dans les groupes Topologiquies et ses Applications.
Paris, Harmann, (1940).
7. Bochner S., Hilbert distance and positive definite functions. Ann. of Math.
42 (1941), p.647-656.
8. Cartan E., Sur la détermination dun système orthgonal complet dans un es-
pace de Riemann symmétrique elos. Rend. Cire. Mat.Palermo, 53 (1929),
p. 217-252.
9. Weil H., Harmonics on homogeneous manifolds. Ann. of Math. 35(1943),
p.486-494.
10. Obukhov A.M., Statistically homogeneous random fields on a sphere. Us-
pehi Mat. Nauk. 2, (1947), p.196-198.
11. Ogura H., Representations of the random fields on a sphere. Mem. Fac.Eng.
Kyoto Univ. 52, N2 (1990), p.81–105.
12. Shoenberg I.J., Positive definite functions on spheres. Duck. Math. J. 9
(1942), p.96–108.
Department of Probability and Mathematical Statistics, Kiev Na-
tional Tarasa Shevchenko University, Kiev, Ukraine
E-mail: probability@mechmat.univ.kiev.ua
Department of Auditing, National Bank of Ukraine, Kiev, Ukraine
E-mail: perun@bank.gov.ua
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