Spectral analysis of multivariate stationary random functions on some massive groups
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 i...
Gespeichert in:
| Datum: | 2007 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут математики НАН України
2007
|
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/4521 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Spectral analysis of multivariate stationary random functions on some massive groups / O. Ponomarenko, Y. Perun // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 177–182. — Бібліогр.: 9 назв.— англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-4521 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-45212009-11-25T12:00:42Z Spectral analysis of multivariate stationary random functions on some massive groups Ponomarenko, O. Perun, Y. 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. 2007 Article Spectral analysis of multivariate stationary random functions on some massive groups / O. Ponomarenko, Y. Perun // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 177–182. — Бібліогр.: 9 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4521 en Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
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. |
| format |
Article |
| author |
Ponomarenko, O. Perun, Y. |
| spellingShingle |
Ponomarenko, O. Perun, Y. Spectral analysis of multivariate stationary random functions on some massive groups |
| author_facet |
Ponomarenko, O. Perun, Y. |
| author_sort |
Ponomarenko, O. |
| title |
Spectral analysis of multivariate stationary random functions on some massive groups |
| title_short |
Spectral analysis of multivariate stationary random functions on some massive groups |
| title_full |
Spectral analysis of multivariate stationary random functions on some massive groups |
| title_fullStr |
Spectral analysis of multivariate stationary random functions on some massive groups |
| title_full_unstemmed |
Spectral analysis of multivariate stationary random functions on some massive groups |
| title_sort |
spectral analysis of multivariate stationary random functions on some massive groups |
| publisher |
Інститут математики НАН України |
| publishDate |
2007 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/4521 |
| citation_txt |
Spectral analysis of multivariate stationary random functions on some massive groups / O. Ponomarenko, Y. Perun // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 177–182. — Бібліогр.: 9 назв.— англ. |
| work_keys_str_mv |
AT ponomarenkoo spectralanalysisofmultivariatestationaryrandomfunctionsonsomemassivegroups AT peruny spectralanalysisofmultivariatestationaryrandomfunctionsonsomemassivegroups |
| first_indexed |
2025-07-02T07:44:48Z |
| last_indexed |
2025-07-02T07:44:48Z |
| _version_ |
1836520349282336768 |
| fulltext |
Theory of Stochastic Processes
Vol.13 (29), no.4, 2007, pp.177–182
OLEKSANDER PONOMARENKO AND YURIY PERUN
SPECTRAL ANALYSIS OF MULTIVARIATE
STATIONARY RANDOM FUNCTIONS ON SOME
MASSIVE GROUPS
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.
1. Introduction
The spectral theory of stationary random functions of second order on
abelian locally compact groups is well developed. This theory is based
on spectral representations of such random functions and their covariance
functions in one-dimensional and multivariate cases [1]-[3].
Now actual problems are connected with obtaining similar results for
weakly stationary random functions on abelian massive groups, i.e. groups
which are not generally locally compact.
We consider two classes of abelian massive groups in this paper. The
first class is nuclear topological groups, which theory was developed in [4]-
[6]. Second class is related with additive groups of general real vector spaces
and corresponding results of harmonic analysis on such groups [7].
The objects of investigation in this paper are spectral representations
for generalized wide sense stationary random functions in Hilbert space H
on nuclear groups and real vector spaces and for their covariance functions.
Under some assumptions of continuity for stationary random functions these
spectral representations are obtained in the form of Fourier transforms of
orthogonally scattered random measures in H and operator positive finite
measures 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 representation
177
178 OLEKSANDER PONOMARENKO AND YURIY PERUN
2. Wide sense stationary random functions in Hilbert space
H on abelian groups
Let L2(Ω) be Hilbert space of all complex-valued random variables of
second order, which defined on some probability space (Ω,F ,P) and H be
a complex Hilbert space. Then the set L(H, L2(Ω)) of all linear continuous
mappings 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 H
and 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 form
on L(H, L2(Ω)) and [Ξ, Ξ] ∈ B+(H).
If dimH = d < ∞ and {ej}d
j=1 is orthonormal basis in H , then random
element Ξ ∈ L(H, L2(Ω)) may be identified with random vector ξ = {ξj}d
j=1,
with ξj = Ξej and expectation EΞ - with vector Eξ = {Eξj}d
j=1 and covari-
ance operator [Ξ, Ψ], Ψ ∈ L(H, L2(Ω)) - with matrix {Eξjϕk}d
j,k=1, where
ϕk = 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 random
elements Ξt ∈ L(H, L2(Ω)) indexed by t ∈ T .
The mean function mt and covariance function Q(s, t) of Ξt, t ∈ T are
defined by the equalities
mt = EΞt, Q(t, s) = [Ξt, Ξs], t, s ∈ T.
Note that Q is positive definite operator kernel on T , i.e. for all integers
n ∈ N, elements tk ∈ T and vectors xk ∈ H, k = 1, ..n
n∑
k=1
n∑
j=1
(Q(tk, tj)xk|xj) ≥ 0.
Conversely, every positive definite operator kernel Q on T with values in
B(H) is a covariance function of some (in particulary gaussian) generalized
random 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 179
Definition 2.2 A random function of second order Ξg, g ∈ G on group G
in H is called a wide-sense (or weakly) stationary if its mean function is
constant mg = EΞg = m, g ∈ G and its covariance function Q(g, s) depends
only 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 stationary
function Ξg, g ∈ G.
Note that definition of stationarity for random function Ξg, g ∈ G is
equivalent to assumption of shift invariance its two moment function mg
and Q(g, s).
3. Spectral representations for stationary random functions
in H on nuclear group
Let V be a real vector space and A, B be non-empty balanced subsets
of V such that A is absorbed by B. For vector subspace Y of V we set
d(A, B; Y ) = inf{t > 0 : A ⊆ tB + Y }.
The Kolmogorov diameter dk of A with respect to B, k ∈ N is defined by
the equality
dk(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 balanced
zero-neighborhood W there exists a balanced zero-neighborhood W0 such
that 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 vector
group.
Definition 3.2 An Abelian topological group is called nuclear if it is
isomorphic to a Hausdorff quotient group of subgroup of some nuclear vector
group.
Note that the class of nuclear groups is a variety of topological Hausdorff
groups, 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 locally
convex spaces [6]. A topological vector space is a nuclear group if and only
180 OLEKSANDER PONOMARENKO AND YURIY PERUN
if it is a nuclear locally convex space [6]. The further information of nuclear
group 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 on
B(H) with respect to family of linear functionals from B(H) into C of the
form: A → tr(BA), A ∈ B(H), where B ranges through the set of trace
class operators in B(H).
Theorem 3.3 1. Let {Ξg, g ∈ G} be a generalized wide sense stationary
random function on nuclear group G in H, which covariance function R(g)
is continuous in ultra-weak topology on B(H).Then exists uniquely defined
random L(H, L2(Ω))-valued strongly regular measure Φ on (Ĝ, τ) such that
Ξg 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 is
connected 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 of
random variables {Ξgx : g ∈ G, x ∈ H}. Define on set of elements in
K(Ξ) of the form
∑n
j=1 Ξgj
xj , which is dense in K(Ξ), the shift operators
Ũs, s ∈ G by equalities
Ũs(
n∑
j=1
Ξgj
xj) =
n∑
j=1
Ξgj+sxj
It is easy to see that these operators are isometric (because Ξg is sta-
tionary function) and may extended by unique way to unitary operators
Us, 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 unitary
representation is connected with stationary function Ξg and its covariance
function R(g) by the equalities
Ξg = UgΞ0, Rg = Ξ∗
0UgΞ0, g ∈ G, (5)
RANDOM FIELDS WITH ISOTROPIC PROPERTY 181
where 0 is neutral element in G and Ξ∗
0 : L2(Ω) → H is adjoint operator for
Ξ0, and also is ultra-weak continuous.
Then from the theorem 13.3 in [6] and theorem 15.4 in [7] it follows
that unitary representation Ug, g ∈ G is the Fourier transform of uniquely
defined strongly regular Radon spectral measure P on (Ĝ, τ) with values
in 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 its
compact subsets, than the assumption of ultra-weak continuity of covariance
function R(g) in theorem 3.3 is equivalent to condition of weak continuity of
R(G) in B(H). In particular, this is true for locally compact or metrizable
groups G.
The result of Remark 3.4 follows from corollary 1.9 and Remark 15.5
from [7].
4. Spectral representations for stationary random function
in H on vector space
For 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 ∈ V
of stationary random function Ξν , ν ∈ V in H is ultra-weakly continuous on
each 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 spectral
representation 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-
tion
Ξν =
∫
V ∗
exp{iλ(ν)}Φ(dλ), ν ∈ V, (7)
where Φ is random L(H, L2(Ω)) -valued measure on (V ∗, Σ(V ∗)) with the
following orthogonal property
[Φ(Δ1), Φ(Δ2)] = F (Δ1 ∩ Δ2). (8)
182 OLEKSANDER PONOMARENKO AND YURIY PERUN
Proof. 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. This
method is based on spectral expansion of unitary representation of additive
group V ,which connected with stationary random function Ξν , ν ∈ V . We
choose here another short way.
First of all we use the fact that R(ν) is positive definite B(H)-valued
function on V . Then through the generalized Bochner theorem for vector
space (Theorem 15.6 from [7]) the spectral representation (6) is valid and
we have the equality
R(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) about
integral spectral representations of generalized random function of second
order in vector space we have representation (7) with property (8).
References
1. Ponomarenko, A.I., Stochastic problems of optimization (in Russian). Kiev
University Press, Kiev (1980).
2. Kakihara Y., Multidimensional second order stochastic processes. World
Scientific, Singapore, New Jersey, London, Hong Kong, (1997).
3. Ponomarenko O.I., To spectral theory of infinite-dimensional homogeneous
in 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 groups
and to the Theory of nuclear groups., Diss. Math. 389 (1999).
5. Auβbenhofer L., A survey on nuclear groups.pp. 1-30, in: Nuclear groups
and Lie groups. Heldermann Verlag Lemgo (2001).
6. Banaszczyk W., Additive Subgroups of Topological Vector Space. Lecture
Notes in Math., 1466, Springer Verlag, Berlin (1991).
7. Glöckner H., Positive definite functions on infinite-dimensional convex
cones. Mem. of AMS, no. 789, vol 166 (2003).
8. Ponomarenko O.I., Random linear functionals of Second Order I. Theory
Probab. and Math. Stat., N54, (1997), p.145–154.
9. Ponomarenko O.I., Integral representation of random functions with values
in 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, Ukraine
E-mail: probability@mechmat.univ.kiev.ua
Department of Auditing, National Bank of Ukraine, Kiev, Ukraine
E-mail: perun@bank.gov.ua
|