The spectral representations for wide sense stationary multivariate random functions and for their covariance functions on two classes of additive vector groups are obtained under some assumptions about continuity of such functions. The ﬁrst class is nuclear topological groups and the second class is additive group of real vector space equipped with the ﬁnite topology

Perun, Y.

Ponomarenko, O.

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

Theory of Stochastic ProcessesVol.13 (29), no.4, 2007, pp.177–182OLEKSANDER PONOMARENKO AND YURIY PERUNSPECTRAL ANALYSIS OF MULTIVARIATESTATIONARY RANDOM FUNCTIONS ON SOMEMASSIVE GROUPSThe spectral representations for wide sense stationary multivariaterandom functions and for their covariance functions on two classes ofadditive vector groups are obtained under some assumptions aboutcontinuity of such functions. The first class is nuclear topologicalgroups and the second class is additive group of real vector spaceequipped with the finite topology.1. IntroductionThe spectral theory of stationary random functions of second order onabelian locally compact groups is well developed. This theory is basedon spectral representations of such random functions and their covariancefunctions in one-dimensional and multivariate cases [1]-[3].Now actual problems are connected with obtaining similar results forweakly stationary random functions on abelian massive groups, i.e. groupswhich are not generally locally compact.We consider two classes of abelian massive groups in this paper. Thefirst class is nuclear topological groups, which theory was developed in [4]-[6]. Second class is related with additive groups of general real vector spacesand corresponding results of harmonic analysis on such groups [7].The objects of investigation in this paper are spectral representationsfor generalized wide sense stationary random functions in Hilbert space Hon nuclear groups and real vector spaces and for their covariance functions.Under some assumptions of continuity for stationary random functions thesespectral representations are obtained in the form of Fourier transforms oforthogonally scattered random measures in H and operator positive finitemeasures in H respectively.2000 Mathematics Subject Classifications. Primary 60G10, 60G57.Key words and phrases. Stationary random functions in Hilbert space,nuclear group,real vector space, spectral representation177178 OLEKSANDER PONOMARENKO AND YURIY PERUN2. Wide sense stationary random functions in Hilbert spaceH on abelian groupsLet L2(Ω) be Hilbert space of all complex-valued random variables ofsecond order, which defined on some probability space (Ω,F ,P) and H bea complex Hilbert space. Then the set L(H, L2(Ω)) of all linear continuousmappings of the space H into L2(Ω) may be considered as the set of gener-alized random elements of second order in H , which defined on (Ω,F ,P).Such generalized random elements may be realized as usual random ele-ments in some quasinuclear extension H− of the space H (see [3], [8]).Denote by B(H) Banach algebra of all linear bounded operators in Hand denote by B+(H) the convex cone of all nonnegative Hermitian opera-tors in B(H).The expectation m = EΞ ∈ H and covariance operator [Ξ, Ψ] ∈ B(H)for generalized random elements Ξ, Ψ ∈ L(H, L2(Ω)) are uniquely determi-nated by the relations(x|m) = E(Ξx), ([Ξ, Ψ]x|y) = E(Ξx)(Ψy), x, y ∈ H,where (·|·) is an inner product in H . Note that [Ξ, Ψ] is sesquilinear formon L(H, L2(Ω)) and [Ξ, Ξ] ∈ B+(H).If dimH = d < ∞ and {ej}dj=1 is orthonormal basis in H , then randomelement Ξ ∈ L(H, L2(Ω)) may be identified with random vector ξ = {ξj}dj=1,with ξj = Ξej and expectation EΞ - with vector Eξ = {Eξj}dj=1 and covari-ance operator [Ξ, Ψ], Ψ ∈ L(H, L2(Ω)) - with matrix {Eξjϕk}dj,k=1, whereϕk = Ψek, k = 1, .., d.Definition 2.1 The (generalized) random function of second order Ξt, t ∈T , in a Hilbert space H defined on set T is the family of generalized randomelements Ξt ∈ L(H, L2(Ω)) indexed by t ∈ T .The mean function mt and covariance function Q(s, t) of Ξt, t ∈ T aredefined by the equalitiesmt = EΞt, Q(t, s) = [Ξt, Ξs], t, s ∈ T.Note that Q is positive definite operator kernel on T , i.e. for all integersn ∈ N, elements tk ∈ T and vectors xk ∈ H, k = 1, ..nn∑k=1n∑j=1(Q(tk, tj)xk|xj) ≥ 0.Conversely, every positive definite operator kernel Q on T with values inB(H) is a covariance function of some (in particulary gaussian) generalizedrandom function of second order Ξt, t ∈ T (see [3]).Let G be some Abelian group with operation which is written as addi-tion.RANDOM FIELDS WITH ISOTROPIC PROPERTY 179Definition 2.2 A random function of second order Ξg, g ∈ G on group Gin H is called a wide-sense (or weakly) stationary if its mean function isconstant mg = EΞg = m, g ∈ G and its covariance function Q(g, s) dependsonly on difference g − s:Q(g, s) = R(g − s), g, s ∈ G. (1)The function R(g) in (1) is called a covariance function of stationaryfunction Ξg, g ∈ G.Note that definition of stationarity for random function Ξg, g ∈ G isequivalent to assumption of shift invariance its two moment function mgand Q(g, s).3. Spectral representations for stationary random functionsin H on nuclear groupLet V be a real vector space and A, B be non-empty balanced subsetsof V such that A is absorbed by B. For vector subspace Y of V we setd(A, B; Y ) = inf{t > 0 : A ⊆ tB + Y }.The Kolmogorov diameter dk of A with respect to B, k ∈ N is defined bythe equalitydk(A, B) = inf {d(A, B; X) : X},where X ranges through the vector subspaces of V with dim(X) < k.Definition 3.1 A locally convex vector group is a real vector space V ,equipped with a Hausdorff group topology such that the filter of zero-neighborhoods possesses a base of symmetric convex sets.A locally convex vector group V is called nuclear if for every balancedzero-neighborhood W there exists a balanced zero-neighborhood W0 suchthat W0 is absorbed by W and for all k ∈ N dk(W0, W ) ≤ k−1.Note that that every locally convex space is a locally convex vectorgroup.Definition 3.2 An Abelian topological group is called nuclear if it isisomorphic to a Hausdorff quotient group of subgroup of some nuclear vectorgroup.Note that the class of nuclear groups is a variety of topological Hausdorffgroups, i.e., is closed under the formation of cartesian products, subgroups,Hausdorff quotients, and passage to isomorphic topological groups; it com-prises the classes of locally compact Abelian groups and nuclear locallyconvex spaces [6]. A topological vector space is a nuclear group if and only180 OLEKSANDER PONOMARENKO AND YURIY PERUNif it is a nuclear locally convex space [6]. The further information of nucleargroup is contained in [4]-[6].Let G be a nuclear group with operation, which is written as addition,and with dual group Ĝ, equipped by admissible topology τ , and for g ∈ G〈χ, g〉 denotes the value of character χ ∈ Ĝ on g.The ultra-weak operator topology on B(H) is the initial topology onB(H) with respect to family of linear functionals from B(H) into C of theform: A 	→ tr(BA), A ∈ B(H), where B ranges through the set of traceclass operators in B(H).Theorem 3.3 1. Let {Ξg, g ∈ G} be a generalized wide sense stationaryrandom function on nuclear group G in H, which covariance function R(g)is continuous in ultra-weak topology on B(H).Then exists uniquely definedrandom L(H, L2(Ω))-valued strongly regular measure Φ on (Ĝ, τ) such thatΞg is the Fourier transform of Φ:Ξg =∫G〈χ, g〉Φ(dχ), g ∈ G. (2)2.The covariance function R(g) of the stationary random function Ξg ad-mits the spectral representation in form of the Fourier transform of B+(H)-valued finite strongly regular operator measure F on (Ĝ, τ):R(g) =∫G〈χ, g〉F (dχ), g ∈ G. (3)The random spectral measure Φ of stationary random function Ξg isconnected with its spectral measure F by the following relation:[Φ(Δ1), Φ(Δ2)] = F (Δ1 ∩ Δ2) (4)Proof. Denote by K(Ξ) the subspace of L2(Ω) generated by family ofrandom variables {Ξgx : g ∈ G, x ∈ H}. Define on set of elements inK(Ξ) of the form∑nj=1 Ξgjxj , which is dense in K(Ξ), the shift operatorsŨs, s ∈ G by equalitiesŨs(n∑j=1Ξgjxj) =n∑j=1Ξgj+sxjIt is easy to see that these operators are isometric (because Ξg is sta-tionary function) and may extended by unique way to unitary operatorsUs, s ∈ G on K(Ξ). It is follows from the definition that the family{Us, s ∈ G} forms unitary representation of group G in K(Ξ). These unitaryrepresentation is connected with stationary function Ξg and its covariancefunction R(g) by the equalitiesΞg = UgΞ0, Rg = Ξ∗0UgΞ0, g ∈ G, (5)RANDOM FIELDS WITH ISOTROPIC PROPERTY 181where 0 is neutral element in G and Ξ∗0 : L2(Ω) → H is adjoint operator forΞ0, and also is ultra-weak continuous.Then from the theorem 13.3 in [6] and theorem 15.4 in [7] it followsthat unitary representation Ug, g ∈ G is the Fourier transform of uniquelydefined strongly regular Radon spectral measure P on (Ĝ, τ) with valuesin B(K(Ξ)). Now spectral representations (2),(3) and property of orthogo-nality (4) are consequences of equalities (5) with Φ(Δ) = P (Δ)Ξ0, F (Δ) =Ξ∗0P (Δ)Ξo.Remark 3.4. If group G is k-space, i.e., if it is the direct limit of itscompact subsets, than the assumption of ultra-weak continuity of covariancefunction R(g) in theorem 3.3 is equivalent to condition of weak continuity ofR(G) in B(H). In particular, this is true for locally compact or metrizablegroups G.The result of Remark 3.4 follows from corollary 1.9 and Remark 15.5from [7].4. Spectral representations for stationary random functionin H on vector spaceFor real vector space, equipped with its finest locally convex topology,denote by V ∗ its topological dual space, equipped with weak -*-topology.Let Σ(V ∗) the coarsest σ-algebra on V ∗, which makes the point evalua-tions λ 	→ λ(ν), λ ∈ V ∗ measurable for all vector ν ∈ V .Let us consider generalized wide sense stationary random function Ξν ,ν ∈ V in Hilbert space H with covariance function R(ν−u) = [Ξν , Ξu], ν, u ∈V .Theorem 4.1. Suppose that B(H)-valued covariance function R(ν), ν ∈ Vof stationary random function Ξν , ν ∈ V in H is ultra-weakly continuous oneach finite-dimensional subspace Y of space V . Then it exists such B+(H)-valued operator measure F on (V ∗, Σ(V ∗)) that R(g) admits the spectralrepresentation in the form of the Fourier transform of measure F :R(ν) =∫V ∗exp{iλ(ν)}F (dλ), ν ∈ V, (6)Itself stationary random function Ξν , ν ∈ V admits the spectral representa-tionΞν =∫V ∗exp{iλ(ν)}Φ(dλ), ν ∈ V, (7)where Φ is random L(H, L2(Ω)) -valued measure on (V∗, Σ(V ∗)) with thefollowing orthogonal property[Φ(Δ1), Φ(Δ2)] = F (Δ1 ∩ Δ2). (8)182 OLEKSANDER PONOMARENKO AND YURIY PERUNProof. One possible way to prove this theorem - it is using the method,which is similar to the approach of proving of theorem 3.4 above. Thismethod is based on spectral expansion of unitary representation of additivegroup V ,which connected with stationary random function Ξν , ν ∈ V . Wechoose here another short way.First of all we use the fact that R(ν) is positive definite B(H)-valuedfunction on V . Then through the generalized Bochner theorem for vectorspace (Theorem 15.6 from [7]) the spectral representation (6) is valid andwe have the equalityR(ν − u) =∫V ∗exp{iλ(ν)}exp{iλ(u)}F (dλ), ν, u ∈ V. (9)Now by mean of application to expansion (9)Theorem 3 from (9) aboutintegral spectral representations of generalized random function of secondorder in vector space we have representation (7) with property (8).References1. Ponomarenko, A.I., Stochastic problems of optimization (in Russian). KievUniversity Press, Kiev (1980).2. Kakihara Y., Multidimensional second order stochastic processes. WorldScientific, Singapore, New Jersey, London, Hong Kong, (1997).3. Ponomarenko O.I., To spectral theory of infinite-dimensional homogeneousin wide sense random fields on groups. Vysnik of Kiev Univ., sec. math.and mech., N11, (1969), p.144–121.4. Auβenhofer L., Contributions to the Duality theory on Topological groupsand to the Theory of nuclear groups., Diss. Math. 389 (1999).5. Auβbenhofer L., A survey on nuclear groups.pp. 1-30, in: Nuclear groupsand Lie groups. Heldermann Verlag Lemgo (2001).6. Banaszczyk W., Additive Subgroups of Topological Vector Space. LectureNotes in Math., 1466, Springer Verlag, Berlin (1991).7. Glöckner H., Positive definite functions on infinite-dimensional convexcones. Mem. of AMS, no. 789, vol 166 (2003).8. Ponomarenko O.I., Random linear functionals of Second Order I. TheoryProbab. and Math. Stat., N54, (1997), p.145–154.9. Ponomarenko O.I., Integral representation of random functions with valuesin locally convex spaces. Theory Probab. and Math. Stat., N46, (1992),p.132–141.Department of Probability and Mathematical Statistics, Kiev Na-tional Tarasa Shevchenko University, Kiev, UkraineE-mail: probability@mechmat.univ.kiev.uaDepartment of Auditing, National Bank of Ukraine, Kiev, UkraineE-mail: perun@bank.gov.ua

Spectral analysis of multivariate stationary random functions on some massive groups

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Spectral analysis of multivariate stationary random functions on some massive groups

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