GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 5, 1998, 415-424

ON A CLASS OF COVARIANCE OPERATORS

T. CHANTLADZE AND N. KANDELAKI

Abstract. This paper is the continuation of [1] in which complex symmetries of distributions and their covariance operators are investigated. Here we also study the most general quaternion symmetries of random vectors. Complete classification theorems on these symmetries are proved in terms of covariance operator spectra.

Let ξ = (ξ′, ξ′′) be a random vector with components from the separable real Hilbert space H_{ℝ} with a scalar product (·|·)_{ℝ}. Thus ξ takes values from the direct sum H_{0} = H_{ℝ} ⊕ H_{ℝ} whose scalar product has the form

(h|g)_{0} = (h′|g′)_{ℝ} + (h′′|g′′)_{ℝ} ∀ h = (h′|h′′), g = (g′, g′′) ∈ H_{0}.

Each element of the space L_{0} = L(H_{0}) of linear bounded operators on H_{0} can be represented as a block matrix

A = (a_{ij})_{i,j=1}^{2} = (\begin{matrix}a_{11}, & a_{12} \\ a_{21}, & a_{22} \end{matrix}) , a_{ij} ∈ L(H_{ℝ}), i, j = 1, 2.  (1)

Let us consider the operator U(x′, x′′) = (−x′′, x′) ∀(x′, x′′) ∈ H_{0} whose representation (1) can be written as

U = (\begin{matrix}0, & −I \\ I, & 0 \end{matrix}) ,  (2)

where I is the unit operator from L(H_{ℝ}). The operator U possesses the important property U^{∗}U = UU^{∗} = I, implying that U is orthogonal and

U^{2} = −I.  (3)

Definition 1. A random vector ξ is called symmetric if (ξ′, ξ′′) and (−ξ′′, ξ′) are equally distributed, i.e., if ξ and Uξ are such.

415 Clearly, if ξ is symmetric, then ξ, −ξ, Uξ, −Uξ are equally distributed.

1991 Mathematics Subject Classification. 60B11, 60B05, 60A10.

Key words and phrases. Random vectors with values from a Hilbert space, covariance operator, proper covariances and distributions, Gaussian distributions, complex and quaternion Hilbert spaces.

1072-947X/98/0900-0415$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1998 Plenum Publishing Corporation

Proposition 1. Let ξ_{1}, ξ_{2}, ξ_{2}, ξ_{4} be the independent copies of a random vector ξ. Then a random vector

ξ_{a} = \frac{1}{2}[ξ_{1} − ξ_{2} + Uξ_{3} − Uξ_{4}]

is symmetric.

The proof immediately follows from (3).

Definition 2. ξ_{a} is called an additive symmetrization of a random vector ξ.

Proposition 2. Let 𝔐 be a random operator taking the values I, −I, U, −U, each with probability \frac{1}{4}. Then a random vector

ξ_{m} = 𝔐ξ

is symmetric if ξ and 𝔐 are independent.

Proof. It follows from (3) that the operator U𝔐 has the same distribution as 𝔐, and does not depend on ξ, so that ξ_{m} and Uξ_{m} are equally distributed.

Definition 3. ξ_{m} is called a multiplicative symmetrization.

Proposition 3. Let a symmetric random vector ξ have a strong moment of second order. Then it is centered and its covariance K_{0} satisfies the relation

UK_{0} = K_{0}U.  (4)

Proof. Since ξ and −ξ are equally distributed, ξ is centered. Since ξ and Uξ are equally distributed, we have K_{0} = U^{∗}K_{0}U. But in that case (3) implies (4).

Denote by L_{1} the commutant of the operator U, i.e., L_{1} = {A|A ∈ L_{0}, AU = UA}. Then (4) implies that K_{0} ∈ L_{1}.

Proposition 4. The operator A belongs to L_{1} if and only if its matrix representation has the form

A = (\begin{matrix}a, & −b \\ b, & a \end{matrix}) , a, b ∈ L(H_{ℝ}).  (5)

The proof follows immediately.

In [1], operators of form (5) are called proper and, accordingly, the centered random vector ξ possessing the covariance K_{0} is called proper for K_{0} ∈ L_{1}, i.e., for

K_{0} = (\begin{matrix}R & −T \\ T & R \end{matrix} ) .  (6)

Generally speaking, if ξ = (ξ′, ξ′′) is the centered random vector with covariance K_{0}, then K_{0} has the form

K_{0} = (\begin{matrix}R_{1} & T^{∗} \\ T & R_{2} \end{matrix}) ,  (7)

where R_{1} and R_{2} are the covariances of ξ′ and ξ′′, respectively, and T is their mutual covariance. Thus the properity of ξ implies that

R_{1} = R_{2} = R, T^{∗} = −T.  (8)

Definition 4. For A ∈ L_{0} we call the expression

A^{_{∣}−_{∣}} = \frac{1}{2}[A + U^{∗}AU]

the averaging of the operator A.

It is easy to verify that the block representation of the averaging K_{0}^{_{∣}−_{∣}} of the operator K_{0} from (7) is written as

K_{0}^{_{∣}−_{∣}} = \frac{1}{2} (\begin{matrix}R_{1} + R_{2} & T^{∗} − T \\ T − T^{∗} & R_{1} + R_{2} \end{matrix}) .

Proposition 5. The operator A ∈ L_{0} is proper, i.e., belongs to L_{1} if

and only if A^{_{∣}−_{∣}} = A.

The proof is obvious.

Let ξ be a centered random vector with covariance K_{0}. Denote by K_{a} and K_{m} the covariances of additive and multiplicative symmetrizations of ξ.

Proposition 6. The equalities

K_{a} = K_{m} = K_{0}^{_{∣}−_{∣}}

are valid.

Proof. The equality K_{a} = K_{0}^{_{∣}−_{∣}} immediately follows from Proposition 1. Indeed, since ξ_{a} = \frac{1}{2}(ξ_{1} − ξ_{2} + Uξ_{3} − Uξ_{4}), we have

K_{a} = \frac{1}{4}(K_{0} + K_{0} + U^{∗}K_{0}U + U^{∗}K_{0}U) = \frac{1}{2}(K_{0} + U^{∗}KU) = K_{0}^{_{∣}−_{∣}}.

Let us prove the equality K_{m} = K_{0}^{_{∣}−_{∣}}. We have

(K_{m}h|h)_{0} = E(ξ_{m}|h)_{0}^{2} = E(𝔐ξ|h)_{0}^{2} = E(ξ|𝔐^{∗}h)_{0}^{2} = E_{𝔐}E_{ξ}(ξ|𝔐^{∗}h)_{0}^{2} =

= E_{m}(K_{0}𝔐^{∗}h|𝔐^{∗}h)_{0} = \frac{1}{4}((K_{0}h|h)_{0} + (K_{0}(−h)|(−h))_{0} +

+ (K_{0}U^{∗}h|U^{∗}h)_{0} + (K_{0}(−U^{∗}h)|(−U^{∗}h))_{0}) =

= \frac{1}{2}((K_{0}h|h)_{0} + (UK_{0}U^{∗}h|h)) = \frac{1}{2}((K_{0} + U^{∗}K_{0}U)h|h)_{0}.

The latter equality follows from (3), which gives U^{∗} = −U.

As shown above (Proposition 4), a symmetric random vector ξ with covariance is proper. The converse statement, generally speaking, does not hold, but we have

Proposition 7. A random proper Gaussian vector is symmetric.

Proof. For the characteristic functional of a proper Gaussian vector we have

φ(Ut) = e^{− \frac{1}{2} (K_{0}Ut|Ut)_{0}} = e^{− \frac{1}{2} (U^{∗}K_{0}Ut|t)_{0}} =
= e^{− \frac{1}{2} (U^{∗}UK_{0}t|t)_{0}} = e^{− \frac{1}{2} (K_{0}t|t)_{0}} = φ(t),

from which it follows that ξ and Uξ are equally distributed, i.e., ξ is symmetric.

Thus if ξ is a proper Gaussian vector, then ξ and ξ_{a} are equally distributed. Moreover, we have

Theorem 1. The distributions of ξ and ξ_{a} coincide if and only if ξ is a random Gaussian vector.

This theorem is a corollary of a more general theorem proved by N. Vakhania in [2].

The averaging K_{0}^{_{∣}−_{∣}} of the covariance operator K_{0} naturally appears as the covariance operator of ξ from some complex Hilbert space H_{1}. Indeed, let us define the multiplication of vectors x ∈ H_{0} by scalars λ ∈ ℂ as

λx = (λ′ + iλ′′)x = (λ′I + λ′′U)x ∀ x ∈ H_{0}, λ ∈ ℂ  (9)

and its Hermitian form as

(x|y)_{1} = (x|y)_{0} + i(x|Uy)_{0} ∀ x, y ∈ H_{0}.  (10)

As a result, the real space H_{0} becomes the complex Hilbert space H_{1} with the Hermitian form (·|·)_{1} and algebra of ℂ-linear operators L_{1}. Hence our random vector ξ becomes the complex random vector ξ = ξ′ + iξ′′, which by virtue of (9) implies that ξ = (ξ′, 0) + U(ξ′′, 0), and its covariance K_{1} with respect to the Hermitian form (·|·)_{1} is defined by the relation

(K_{1}h|h)_{1} = \frac{1}{2}E|(ξ|h)_{1}|^{2}.  (11)

Proposition 8. The equality

K_{1} = K_{0}^{_{∣}−_{∣}}

is valid.

Proof.

(K_{1}h|h)_{1} = \frac{1}{2}E|(ξ|h)_{1}|^{2} = \frac{1}{2}E((ξ|h)_{0}^{2} + (ξ|Uh)_{0}^{2}) = \frac{1}{2}(K_{0}h|h)_{0} +

+ \frac{1}{2}(U^{∗}K_{0}Uh|h)_{0} = (\frac{1}{2}(K_{0} + U^{∗}K_{0}U)h|h)_{0} = (K_{0}^{_{∣}−_{∣}}h|h)_{0}.

Since by (11) (K_{1}h|h)_{1} is real, by (10) (K_{1}h|h)_{1} = (K_{1}h|h)_{0} and therefore

K_{1} = K_{0}^{_{∣}−_{∣}}.

This important equality is thoroughly investigated in [1].

The above reasoning can be summarized as

Theorem 2. The following conditions are equivalent: (a) ξ is a proper

random vector (see (6) and (8)); (b) K_{0} = K_{0}^{_{∣}−_{∣}}; (c) K_{0} = K_{a}; (d) K_{0} = K_{m};

(e) K_{0} = K_{1}.

Let ξ be centered and have the covariance K_{0}. Recall that K_{0} is a completely continuous, positive-definite operator. Denote by M_{i} (i = 1, 2, …) the eigensubspaces of K_{0} which correspond to the eigenvalues μ_{i}, (i = 1, 2, …), μ_{1} > μ_{2} > ⋯; μ_{i} → 0. If the operator K_{0} is degenerate, denote M_{0} = Ker K_{0}. Let us now describe proper covariance operators in terms of their eigenvalues.

Definition 5. A random vector ξ is called orthogonally proper if there exists an orthogonal operator O such that Oξ is a proper random vector.

Theorem 3. A random vector ξ is orthogonally proper if and only if dim M_{k} is even for all k ≥ 0.

Proof. Sufficiency. Let dim M_{k} be even for all k ≥ 0. Then there exists a basis of the space H_{0} consisting of eigenvectors of the operator K_{0} : f_{1}, f_{2}, … such that for K_{0}f_{2p−1} = μf_{2p−1} we have K_{0}f_{2p} = μf_{2p}, p = 1, 2, . . .. Let further g_{p} (p = 1, 2, …) be a basis of H_{ℝ}. Then a sequence of vectors e_{1}, e_{2}, … from H_{0}, for which e_{2p−1} = (g_{p}, 0), e_{2p} = (0, g_{p}), p = 1, 2, …, is the orthonormal basis H_{0}. By (2) Ue_{2p−1} = e_{2p}, Ue_{2p} = −e_{2p−1} (p = 1, 2, …). Let us now define the orthogonal operator O as Oe_{k} = f_{k} (k = 1, 2, …). The covariance Oξ has the form Γ = O^{−1}K_{0}O. Next we shall prove that ΓU = UΓ. Indeed, we have

UΓe_{2p−1} = UO^{−1}K_{0}Oe_{2p−1} = UO^{−1}K_{0}f_{2p−1} = UO^{−1}μ_{k}f_{2p−1} =
= μ_{k}UO^{−1}f_{2p−1} = μ_{k}e_{2p−1} = μ_{k}e_{2p};
ΓUe_{2p−1} = O^{−1}K_{0}e_{2p} = O^{−1}K_{0}f_{2p} = μ_{k}O^{−1}f_{2p} = μ_{k}e_{2p},

p = 1, 2, . . .. The same reasoning is true for e_{2p} (p = 1, 2, …).

Necessity. Let O be an orthogonal operator such that Oξ is proper. Then M′_{k} = O^{−1}M_{k} (k ≥ 0) are the eigensubspaces of Γ and μ_{k} are the corresponding eigenvalues. Indeed, if x ∈ M_{k}, then Γx = O^{−1}K_{0}Ox. Since Ox ∈ M_{k}, we have Γx = O^{−1}μ_{k}Ox = μ_{k}x. Conversely, assume that Γx = μ_{k}x. Then, on the one hand, μ_{k}x = Γx = O^{−1}(K_{0}Ox), and, on the other, O^{−1}(μ_{k}Ox) = μ_{k}x. Hence it follows that K_{0}Ox = μ_{k}Ox and therefore Ox ∈ M_{k}, i.e., x ∈ M′_{k}. Since Γ is a proper operator, we have ΓU = UΓ. Thus we conclude that M′_{k} are invariant for U as well. Indeed, assuming that x ∈ M′_{k}, we obtain ΓUx = UΓx = Uμ_{k}x = μ_{k}Ux. But this implies that, together with x, Ux also belongs to M′_{k}. But this is possible only when dim M′_{k} is even. Therefore dim M_{k} will be even too. (Throughout the paper it is assumed that “infinity” may have any multiplicity).

Remark 1. The orthogonal property is understood as a property with respect to U_{1} = OUO^{−1} in the sense that U_{1} is orthogonal, U_{1}^{2} = −I, and U_{1}K_{0} = K_{0}U_{1}.

Since K_{0} is the self-conjugate operator, H_{0} can be represented as an orthogonal sum H_{0} = \overline{im K_{0}} + Ker K_{0}. e_{1}, e_{2}, … is the orthonormal basis of the subspace \overline{im K_{0}} consisting of the eigenvectors of K_{0} corresponding to the values λ_{1} ≥ λ_{2} ≥ · · ·.

Theorem 4. A self-conjugate invertible operator S, permutable with K_{0} and such that a random vector η = Sξ is orthogonally proper, exists if and only if inf_{p≥1} \frac{λ_{2p}}{λ_{2p−1}} = c_{0} > 0 and Ker K_{0} is even.

Proof. Sufficiency. We write S as S_{e_{k}} = σ_{k}e_{k}, S_{x} = x ∀x ∈ Ker K_{0}, where σ_{k} are defined by the equalities σ_{2p} = 1, σ_{2p−1} = (λ_{2p}/λ_{2p−1})^{1/2} p = 1, 2, . . .. It is obvious that S is permutable with K_{0} and self-conjugate. We shall prove that it is invertible. Note that \sqrt{c_{0}} ≤ σ_{k} ≤ 1 for all x.

The right-hand side of the inequality follows from the definition of σ_{k} and from λ_{2p} ≤ λ_{2p−1} (p = 1, 2, …), while the left-hand side is provided by the condition of the theorem inf_{p≥1} \frac{λ_{2p}}{λ2p−1} = c_{0} > 0. Therefore S is invertible. Finally, let us prove that the random vector η = Sξ is orthogonally proper. Since S is self-conjugate and permutable with K_{0}, the covariance η is S^{∗}K_{0}S, we obtain S^{∗}K_{0}S = S^{∗}SK_{0} = S^{2}K_{0}. Thus the covariance η in the basis e_{1}, e_{2}, … can be represented in the diagonal form σ_{1}^{2}λ_{1}, σ_{2}^{2}λ_{2}, . . .. But σ_{2p}^{2}λ_{2p} = λ_{2p}, σ_{2p−1}^{2}λ_{2p−1} = (λ_{2p}/λ_{2p−1})λ_{2p−1} = λ_{2p}, p = 1, 2, . . .. Therefore all eigensubspaces S^{∗}K_{0}S (including Ker K_{0}) are even, i.e., η = Sξ is orthogonally proper.

Necessity. Let S satisfy the conditions of the theorem: S is invertible, self-conjugate, permutable with K_{0}, and the dimensions of the eigensubspaces S^{∗}K_{0}S are even. Hence it immediately follows that Ker K_{0} = Ker S^{∗}K_{0}S and im S^{∗}K_{0}S. Clearly, S_{e_{k}} = σ_{k}e_{k}, k = 1, 2, …, α_{0} ≤ σ_{k} ≤ α_{1}, for all k and σ_{k}^{2}λ_{k} = β_{k} are such that β_{2p−1} = β_{2p} p = 1, 2, . . .. Further, λ_{2p}/λ_{2p−1} = (σ_{2p−1}/σ_{2p})^{2}β_{2p}/β_{2p−1} ≥ (\frac{α_{0}}{α_{1}} )^{2} > 0.

We shall now give an example of a random vector which does not belong to the above-mentioned class. Let ξ be a random vector with covariance K_{0} whose eigenvalues λ_{k} = e^{−k^{α}} (α > 1) (k = 1, 2, …). It is assumed that the conditions of Theorem 4 are fulfilled for α = 1 and for all orders less than e^{−k}.

Proposition 9. The operators from L_{0} = L(H_{0})

U = (\begin{matrix}0, & −I \\ I, & 0 \end{matrix}) , V = (\begin{matrix}v, & 0 \\ 0, & −v \end{matrix}) ,  (12)

where v is an orthogonal operator in H_{ℝ} such that v^{2} = −I (for the existence of such an operator see [3], [4]), give, in H_{0}, an additional structure (see [5], Ch. 11, §2), i.e., U and V are orthogonal in H_{0} and

U^{2} = V^{2} = −I, UV = −V U.  (13)

Proof. Follows immediately.

Theorem 5. The covariance K_{0} of a proper random vector (see (6), (8)) is permutable with V if and only if

Rv = vR, T_{v} = −vT.

Proof. Follows immediately.

Definition 6. A proper random vector ξ is called twice proper if its covariance operator K_{0} is permutable with V.

The operators defined by formulas (12), (13) allow us to introduce, in the space H_{0}, the so-called Hermitian form defining one more covariance K for ξ. We shall see below that for twice proper random vectors K and K_{0} coincide.

Consider the set

ℍ = {A|A ∈ L_{0}A = aI + bU + cV + dW, a, b, c, d ∈ ℝ},

where W = UV. Note that W is an orthogonal operator such that W U = −UW, W V = −V W. It is easy to verify that ℍ is a four-dimensional ℝ-algebra with involution with respect to standard operations A + B, AB, λA, A^{∗}. Moreover, ℍ is a body whose every nonzero element is invertible. Indeed, for any A = aI +bU +cV +dW we have AA^{∗} = (a^{2} +b^{2} +c^{2} +d^{2})I, which means that if a ≠ 0, then A^{−1} exists and A^{−1} = (a^{2}+b^{2}+c^{2}+d^{2})^{−1}A^{∗}. The set ℍ acts on H_{0} in a natural way: if A ∈ ℍ, x ∈ H_{0}, then its action is simply Ax. Finally, the mapping (·|·) : H_{0} ×H_{0} → ℍ defined by the formula

∀(x, y) ∈ H_{0} × H_{0} (x|y) = (x|y)_{0}I + (x|Uy)_{0}U +
+(x|V y)_{0}V + (x|W y)_{0}W  (14)

gives the Hermitian form. Indeed, it is easy to verify that

(x + y|z) = (x|z) + (y|z) ∀ x, y ∈ H_{0},
(Ax|y) = A(x|y) ∀A ∈ ℍ, x, y ∈ H_{0}; (x|y)^{∗} = (y|x) ∀x, y ∈ H_{0},

and, finally, (x|x) ≥ 0 ∀x ∈ ℍ (x|x) = 0 ⇔ x = 0, where A ≥ 0 means that the operator A is nonnegative. But since |(|x)| = (x|x)_{0}, the topologies generated by the Hermitian forms ·|·) and ·|·)_{0} in H_{0} coincide.

For the centered random vector ξ with covariance K_{0} we define the covariance K with respect to the Hermitian form (14) by the formula (Kh|h) = \frac{1}{4}E|(ξ|h)|^{α}.

Proposition 10. The covariance K has the form

K = \frac{1}{4}(K_{0} + U^{∗}K_{0}U + V^{∗}K_{0}V + W^{∗}K_{0}W).  (15)

Proof. (14) implies

K = \frac{1}{4}[E(ξ|h)_{0}^{2} + E(ξ|Uh)_{0}^{2} + E(ξ|V h)_{0}^{2} + E(ξ|W h)_{0}^{2}] =

= \frac{1}{4}[(K_{0}h|h)_{0} + (K_{0}Uh|Uh)_{0} + (K_{0}V h|V h)_{0} + (K_{0}W h|W h)_{0}] =

= \frac{1}{4}((K_{0} + U^{∗}K_{0}U + V^{∗}K_{0}V h + W^{∗}K_{0}W)h|h)_{0}.

Since, by definition, (Kh|h) coincides with (Kh|h_{0})_{0}, we obtain (15).

Proposition 11. If a random vector is twice proper, then

K_{0} = K.

So far we have been concerned with introducing new structures in the space H_{0}. A similar situation occurs in the case of the narrowing of a scalar field when a rich structure is given for the initial space and we are interested in the behavior of a random vector in different structures. The case ℝ ⊂ ℂ is treated in [1]. Here we shall consider the restriction of scalars in the quaternion case. Recall that the quaternion body ℍ is a four-dimensional ℝ-algebra with the basis 1, i, j, k and multiplication table

i j k
i -1 k -j
j -k -1 i
k j -i -1

The conjugate quaternion \overline{μ} to μ = μ′ + iμ′′ + jμ′′′ + kμ^{IV} is defined as \overline{μ} = μ′−iμ′′−jμ′′′−kμ^{IV}. If λ = λ′+iλ′′+jλ′′′+kλ^{IV}, then the expression (λ|μ) = λ\overline{μ} defines the quaternion Hermitian form, while (λ|μ)_{0} = Re λ\overline{μ} gives a Euclidean scalar product ℍ over ℝ^{0} : (λ|μ)_{0} = λ′μ′ +λ′′μ′′ +λ′′′μ′′′ + λ^{IV} μ^{IV} with the norm |λμ|_{0} = |λμ| = |λ|_{0}|μ|_{0}, where, as usual, it is assumed that | · |_{0} = (·|·)_{0}^{1/2}. Note that ℝ ⊂ ℂ ⊂ ℍ. The quaternion Hilbert space H is an Abelian group with respect to the sum x + y of the vectors from H, where the product λx of scalars from ℍ and of scalars from H is defined. Moreover, this product is complete and separable with respect to the following Hermitian form with values in H given on ℍ : ∀x, y ∈ H:

(x|y) = (x|y)_{0} + (x|iy)_{0}i + (x|jy)_{0}j + (x|ky)_{0}k.

Here

(·|·)_{0} = Re(·|·)

is the usual real scalar product. Thus, in the case of the restriction of the scalars from ℍ to ℝ, the space H transforms to the real Hilbert space H_{0}, while in the case of the restriction to ℂ, it transforms to the complex Hilbert space H_{1} with the Hermitian form

( · | · ) = ( · | · )_{0} + ( · |i(·))_{0}i.

Since | · | = | · |_{1} = | · |_{0}, we find that H^{0}, H_{1}, H_{0}, being topological Abelian groups, completely coincide.

Denote by L, L_{1}, L_{0} the spaces of bounded linear operators ℍ, ℂ, ℝ, respectively. The restriction of the scalars gives the natural embeddings L ⊂ L_{1} ⊂ L_{0}. Now we introduce in H_{0} the operators U and V

U = iI, V = jI.

which, as is easy to verify, satisfy (13). Therefore all the above arguments hold for the random vector ξ with values from H_{0}. Thus Theorem 3 will read as follows:

Theorem 6. A random vector ξ is twice orthogonally proper if and only if dim M_{k} is a multiple of four for all k ≥ 0.

Acknowledgment

This study was financially supported by the Soros Foundation.

References

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2. N. Vakhania, On Polya’s characterization theorem for complex random variables. J. Complexity 13(1997), No. 4, 480–488.

3. N. N. Vakhania and N. P. Kandelaki, Orthogonal random vectors in a Hilbert space. (Russian) Trudy Inst. Vychisl. Matem. Akad. Nauk GSSR 28(1988), No. 1, 11–37.

4. N. Vakhania, Orthogonal random vectors and the Hurwitz–Radon– Eckman theorem. Georgian Math. J. 1(1994), No. 1, 99–113.

5. D. Husemoller, Fiber bundles. McGraw-Hill Book Company, New York, 1966.

(Received 01.08.1996)

Authors’ address:

N. Muskhelishvili Institute of Computational Mathematics

Georgian Academy of Sciences

8, Akuri St., Tbilisi 380093

Georgia