# Stochastic Mappings and Random Distribution Fields. A Correlation Approach

Păstorel Gaşpar  Department of Mathematics and Computer Science, Faculty of
Exact Sciences, “Aurel Vlaicu” University, Arad, Str. E. Drăgoi nr. 2, 310330
Arad, Romania<EMAIL_ADDRESS>and  Lorena Popa<EMAIL_ADDRESS>
###### Abstract.

This paper contains a study of multivariate second order stochastic mappings
indexed by an abstract set $\Lambda$ in close connection to their operator
covariance functions.
The characterizations of the normal Hilbert module or of Hilbert spaces
associated to such a multivariate second order stochastic mapping in terms of
reproducing kernel structures are given, aiming not only to gather into a
unified way some concepts from the field, but also to indicate an instrument
for extending the very well elaborated theory of multivariate second order
stochastic processes (or random fields) to the case of multivariate second
order random distribution fields, including multivariate second order
stochastic measures. In particular a general Wold type decomposition is
extended and discussed in our framework.

###### Key words and phrases:

stochastic mapping, correlation function, positive definite kernels,
reproducing kernel Hilbert modules, Kolmogorov factorization, stochastic
measures, random distribution fields

###### 1991 Mathematics Subject Classification:

60G20 ; 47B32

##  1\. Introduction

The study of stochastic or random processes (i.e. a family of random
variables) is nowadays among the important topics in mathematical research.
The family of random variables composing a stochastic process was, naturally,
first indexed by the set of integers ${\mathbb{Z}}$ and then by the set of
real numbers ${\mathbb{R}}$. Further the necessity of simultaneous study of
more stochastic processes or to describe more complicated phenomena from
technical, natural or social sciences, led to consider the process as
consisting of multivariate (finite or infinite dimensional) random variables
or even generalized random variables (i.e. continuous linear operators sending
the complex random variables into some Hilbert space) and, on the other side,
to index the process by ${\mathbb{Z}}^{d}$ and ${\mathbb{R}}^{d}$ (with $d$ a
positive integer) or, even by an abstract semigroup or other algebraic-
topological structures.

In an attempt to gather the existing frameworks for studying random fields we
introduce _multivariate stochastic mappings_ as families of $H$-valued random
variables indexed by an arbitrary set $\Lambda$ (often taken with a
topological structure), with $H$ an infinite dimensional separable complex
Hilbert space.

The plan of the paper runs as follows. In Section 2 the basic concepts
regarding the organization of the set of second order $H$-valued random
variables into a normal Hilbert ${\mathcal{B}}(H)$-module and some other basic
notations are introduced, the outline of the chosen framework and it’s
connections to the developments from the specialized literature being
mentioned.
Section 3 presents basic results regarding the correlation theory (including a
Kolmogorov-type factorization), formulated in the general setting of
multivariate second order stochastic mappings, where reproducing kernel
techniques are essentially used.
In Section 4 multivariate second order random distribution fields are defined
and using the space ${\mathcal{D}}_{d}$ of test functions on
${\mathbb{R}}^{d}$ from distribution theory as index set, the results from
Section 3 are applied to obtain a correlation theory for such type of
stochastic mappings. A general Wold decomposition in this extended framework
ends this Section.
Since, in further developments of the theory of multivariate random
distribution fields their subclass of regular multivariate stochastic not
necessarily bounded measures on ${\mathbb{R}}^{d}$ play an important role,
Section 5 is devoted to transpose the results of Sections 3 and 4 to such
measures as stochastic mappings indexed by some $\delta$-rings of Borel sets
in ${\mathbb{R}}^{d}$.
Finally in Section 6 considerations about next steps in developing the theory
in this framework are made.

##  2\. Preliminaries and basic notations

We start with a probability space $(\Omega,{\mathscr{A}},\wp)$ and denote by
$L^{2}(\wp)$ the Hilbert space of second order complex valued random variables
and by $L^{2}_{s}(\wp,H)$ the Hilbert space of strong second order $H$ \-
valued random variables. More generally, a function $f:\Omega\rightarrow H$ is
from $L^{r}_{s}(\wp,H),\ 1\leq r<\infty$, if it is strongly measurable (in
this case equivalent to weakly measurable) and the $r$th power of its norm is
integrable, the spaces being endowed with the usual $r$-norms. Hence, the
scalar product in $L^{2}_{s}(\wp,H)$ takes the natural form:

(2.1)
$(f,g)_{L_{s}^{2}(\wp,H)}:=\int\limits_{\Omega}\left(f(\omega),g(\omega)\right)_{H}d\wp(\omega)\
;\quad f,g\in L^{2}_{s}(\wp,H)$

(see also [14] pp.11). The space $L^{2}_{w}(\wp,H)$ of weak second order
$H$-valued random variables being in an obvious way defined, it is not hard to
see that $L^{2}_{s}(\wp,H)\subset L^{2}_{w}(\wp,H)\subset L^{1}_{s}(\wp,H)$.
Then we can identify in $L^{2}_{s}(\wp,H)$ and also in $L^{2}_{w}(\wp,H)$ the
subspaces of strong, respectively weak second order $H$-valued random
variables of zero mean 111Since the random variables composing a stochastic
process or random field are supposed to have the same mean, we might as well
consider only the ones of zero mean. $L^{2}_{s,0}(\wp,H)$ and
$L^{2}_{w,0}(\wp,H)$ respectively. Now, the mapping

(2.2) $f\mapsto V_{f}$

where

(2.3) $V_{f}\chi=\int_{\Omega}\chi(\omega)f(\omega)d\wp(\omega),\ \ \ \chi\in
L^{2}_{0}(\wp),$

enables us to regard the elements of these two spaces as Hilbert space
morphisms between $L^{2}_{0}(\wp)$ and $H$. This is done by endowing them with
a structure of inner product ${\mathcal{B}}(H)$-modules. Therefore we
establish the following notations. When $G$ is another complex Hilbert space,
then ${\mathcal{B}}(G,H)$ means the space of continuous linear operators,
while ${\mathscr{C}}_{2}(G,H)$ is the class of Hilbert-Schmidt operators from
$G$ to $H$. ${\mathscr{C}}_{1}(H)$ is the ideal of trace class operators from
the $C^{*}$-algebra ${\mathcal{B}}(H)={\mathcal{B}}(H,H)$. Now,
$L^{2}_{s,0}[\wp,H]$ denotes $L^{2}_{s,0}(\wp,H)$ organized as a normal
Hilbert ${\mathcal{B}}(H)$-module with the natural outer action of
${\mathcal{B}}(H)$ and the ${\mathscr{C}}_{1}(H)$-valued inner product
(Gramian) defined 222here the tensor product is understood in the sense of
Schatten, i.e. $(h\otimes\bar{k})h^{\prime}:\ =(h^{\prime},k)_{H}h;\
h,h^{\prime},k\in H$ by

(2.4) $[f,g]_{L^{2}_{s,0}[\wp,H]}:\ =\int
f(\omega)\otimes\overline{g(\omega)}d\wp(\omega),\ f,g\in L^{2}_{s,0}[\wp,H],$

and the topology given by the norm

(2.5) $\|f\|_{L^{2}_{s,0}[\wp,H]}=\|[f,f]\|^{1/2},\qquad f\in
L^{2}_{s,0}[\wp,H].$

Let’s note that because of the connection
$(f,g)_{L^{2}_{s,0}(\wp,H)}={\mathrm{tr}}[f,g]_{L^{2}_{s,0}[\wp,H]}$ between
the scalar product (2.1) and the Gramian (2.4), the norm (2.5), defined above,
actually coincides with the natural $L^{2}$-norm.

Thus, the mapping (2.2) is a (Gramian preserving) module isomorphism of
$L^{2}_{s,0}[\wp,H]$ onto the normal Hilbert ${\mathcal{B}}(H)$-module
${\mathscr{C}}_{2}(L^{2}_{0}(\wp),H)$, while its extension to
$L^{2}_{w,0}(\wp,H)$ embeds it in a natural way into the Hilbert
${\mathcal{B}}(H)$-module ${\mathcal{B}}(L^{2}_{0}(\wp),H)$.

The elements of ${\mathscr{H}}$, which as in [14] represents henceforth a
common notation for $L^{2}_{s,0}[\wp,H]$ and
${\mathscr{C}}_{2}(L^{2}_{0}(\wp),H)$, respectively the elements of
${\mathcal{B}}(L^{2}_{0}(\wp),H)$ (including those from $L^{2}_{w,0}(\wp,H)$)
will be called _multivariate strong second order random variables of zero
mean_ , respectively _generalized multivariate second order random variables
of zero mean_. In what follows instead of _strong second order_ we use simply
the term _second order_.

Generally speaking a (generalized) multivariate stochastic process is a family
$\\{\Phi_{\lambda}\\}_{\lambda\in\Lambda}$ of (generalized) multivariate
random variables, where if $\Lambda=\mathbb{Z}^{d}$, we have a _d-time
discrete parameter (generalized) multivariate stochastic process_ or, if
$\Lambda=\mathbb{R}^{d}$, we have a _d-time continuous parameter (generalized)
multivariate stochastic process_.
Since, in our study we shall use Hilbert space operator methods,
$\Phi_{\lambda},\linebreak\lambda\in\Lambda$ will be required to be strongly
(or weakly) square integrable and, without restraining the generality, of zero
mean, i.e. $\Phi_{\lambda}\in L^{2}_{s,0}(\wp,H)$ (or $\Phi_{\lambda}\in
L^{2}_{w,0}(\wp,H)\subset{\mathcal{B}}(L^{2}_{0}(\wp),H)$). So, in these cases
we speak about $\\{\Phi_{\lambda}\\}_{\lambda\in\Lambda}$ as being a
_(generalized) multivariate second order stochastic process_. For an embedding
as before, see also [31] or [3], where Banach space valued random variables
are considered and the term of _generalized second order stochastic process_
in this sense first appears.

When $d>1$, or even when $\Lambda$ is a locally compact abelian group, the
term _(generalized) random field_ is preferred instead of _(generalized)
stochastic process_ , while when $\Lambda$ is an arbitrary index set, the term
_(generalized) stochastic mapping_ is the most appropriate.

So, the basic concept which we shall use in this paper will be a function
$\Phi$ defined on an arbitrary set $\Lambda$,

(2.6) $\Lambda\ni\lambda\mapsto\Phi(\lambda)\in{\mathscr{H}}\left(\text{ or
}{\mathcal{B}}(L^{2}_{0}(\wp),H)\right),$

which we shall call (generalized) multivariate second order stochastic
mapping, briefly (g.)m.s.o.s.m.

We shall mention further how several such concepts from the scientific
literature fit in our present setting. When $\Lambda$ is a separable metric
space (which can automatically be regarded as a measurable space with respect
to a positive measure on the $\sigma$-algebra $\Sigma={\mathcal{B}}or\Lambda$
of Borel sets), then we find ourself with multivariate stochastic mappings in
the framework of _infinite dimensional random mappings_ from the paper [25].
Let us mention that in this framework regularity conditions (some kinds of
continuity or measurability) for m.s.o.s.m. occur naturally. They can also be
formulated even in some enlarged frameworks. For instance, if the parameter
set $\Lambda$ is a separate topological space, then we can - obviously -
consider continuous m.s.o.s.m. (a.k.a. stochastically continuous [25]), while
if $\Lambda$ is a measurable space, i.e. it is endowed with a positive measure
$\nu$ on a $\sigma$-algebra $\Sigma$ of subsets of $\Lambda$
($\Lambda=(\Lambda,\Sigma,\nu)$), we may speak about measurable m.s.m. (see
[25, pp. 54]), as well as of (square) summable or locally (square) summable
m.s.o.s.m. (see [15], [5]).

Also covered by this concept are the stochastic or random measures (see [27]),
_stochastic or random integrals_ and the _random operators_ (see [25, pp. 54,
55]). For example, when $\Lambda=H$ and $\Phi$ is a continuous linear operator
from $H$ to $L^{r}_{s}(\wp,H)$, then it is known as a _random bounded
operator_ (see [22] and, for recent results, [25], [26], [27]), whereas if
$\Lambda$ is a dense subspace in $H$, then a closed linear operator from
$\Lambda$ to $L^{r}_{s}(\wp,H)$ is a _random (not necessarily bounded)
operator_ (see [12]).

The concept of univariate second order stochastic mapping was mentioned as
such by H. Niemi in [17, pp. 6], while the concept of the generalized
multivariate second order stochastic mapping was introduced by P. Masani in
[16] under the name _hilbertian variety_. In the framework of this last
concept we may place also the works of W. Hackenbroch [11] on Hilbert space
operator valued processes, S. A. Chobanyan and A. Weron [3] on prediction
theory in Banach spaces and of I. Suciu and I. Valuşescu [23],[28], [29] about
the study of stochastic processes in the context of complete correlated
actions.

An important tool in the development of the theory in all mentioned areas is
the reproducing kernel technique for Hilbert spaces and Hilbert
$C^{*}$-modules which are well presented in [2], respectively in [24].

In what follows we restrain ourselves to the study of _multivariate second
order (m.s.o.) stochastic mappings_ , which cover not only some particular
concepts used in the very well developed theory of _m.s.o. random fields_ (see
[14]), but also an extension which we have in view.

Namely, if we intent to consider the m.s.o. stochastic processes not only as
${\mathscr{H}}$-valued or ${\mathcal{B}}(L^{2}_{0}(\wp),H)$-valued _functions_
on ${\mathbb{R}}^{d}$, but more generally as ${\mathscr{H}}$-valued,
respectively ${\mathcal{B}}(L^{2}_{0}(\wp),H)$-valued _distributions_ on
$\mathbb{R}^{d}$ (see [21]), then it is necessary to have
$\Lambda={\mathcal{D}}({\mathbb{R}}^{d})={\mathcal{D}}_{d}$, the space of test
functions in the theory of distributions. For such a (generalized) m.s.o.
stochastic mapping we shall use the term _(generalized) m.s.o. random
distribution field (m.s.o.r.d.f.)_.

Mentioning that, for the univariate one time parameter case, such an extension
of stochastic processes was considered for the first time by K. Itô ([13]) and
I.M.Gelfand ([8]) in 1953 and 1955 respectively and then for the finite
variate $d$ time parameteres case in 1957 by A. M. Yaglom [35] (see also [4],
[9], [34], [36], [33]), we emphasize that the starting point of our research
was the extension of the theory of m.s.o. random fields on ${\mathbb{R}}^{d}$
(treated in [14]) to the _m.s.o. random distribution fields_.

##  3\. Correlation theory of multivariate second order stochastic mappings

In this section we consider _(continuous) m.s.o. stochastic mappings_ ,
indexed over an abstract set (topological space) $\Lambda$, which in what
follows will be denoted (in both cases) by
$\boldsymbol{M}(\Lambda,{\mathscr{H}})$. In analogy to [14, Section 4.1] we
associate to such stochastic mappings _the vector domain, the modular domain,
the measurements space and the operator and scalar (cross) covariance
functions_. The corresponding modular and vector domains are then
characterized as reproducing kernel structures (a Hilbert module and a Hilbert
space) considered in [14, Section 2.4], reproduced by a positive definite
${\mathscr{C}}_{1}(H)$-valued kernel on $\Lambda$, which is just the operator
covariance function of the stochastic mapping. Analogously the measurements
space will be identified as a Hilbert space reproduced by the scalar
covariance function of the stochastic mapping. Also for a general positive
definite ${\mathscr{C}}_{1}(H)$-valued kernel, as considered in [14, Section
2.4], a Kolmogorov type factorization in terms of a m.s.o. stochastic mapping
is obtained. Thus, this Section can be regarded as being complementary to
Section 2.4. from the book [14].

For a given m.s.o.s.m. $\Phi\in\boldsymbol{M}(\Lambda,{\mathscr{H}})$ we
denote by ${\mathscr{H}}_{(\Phi)}$ the closed linear subspace of
$L_{s,0}^{2}(\wp,H)$ generated by the values $\\{\Phi(\mu),\mu\in\Lambda\\}$
and call it _the vector domain_ of $\Phi$, while the _modular domain_ of
$\Phi$ is the closure ${\mathscr{H}}_{\Phi}$ in $L_{s,0}^{2}(\wp,H)$ of the
submodule

(3.1) ${\mathscr{H}}^{0}_{\Phi}:\
=\left\\{\sum\limits_{i\in\mathbb{N}_{m}}a_{i}\Phi(\lambda_{i})\,,a_{i}\in{\mathcal{B}}(H),\lambda_{i}\in\Lambda,i\in\mathbb{N}_{m},m\in\mathbb{N}\right\\},$

where ${\mathbb{N}}_{m}=\\{1,2,\dots,m\\}$. Obviously,
${\mathscr{H}}_{(\Phi)}\subset{\mathscr{H}}_{\Phi}\subset{\mathscr{H}}$. These
terms can serve also to introduce in $\boldsymbol{M}(\Lambda,{\mathscr{H}})$
the relation of subordination, which will be useful in the general Wold
decomposition. Namely, we shall say that $\Phi$ is _subordinate (operator
subordinate)_ to $\Psi$ if
${\mathscr{H}}_{(\Phi)}\subset{\mathscr{H}}_{(\Psi)}$
(${\mathscr{H}}_{\Phi}\subset{\mathscr{H}}_{\Psi}$, respectively).
Now, using the operator model of the normal Hilbert ${\mathcal{B}}(H)$-module
${\mathscr{H}}_{\Phi}$ (see Corr. 7 pp. 30 of [14]) and denoting
$G_{\Phi}:=G_{{\mathscr{H}}_{\Phi}}$, which will be called the _measurements
space_ of $\Phi$, we have the module isomorphisms

(3.2)
${\mathscr{H}}_{\Phi}\cong\mathscr{C}_{2}(G_{\Phi},H)=H\widehat{\otimes}_{2}\overline{G}_{\Phi}\cong
H\widehat{\otimes}G_{\Phi},$

where $H\widehat{\otimes}G$ is the hilbertian tensor product of the Hilbert
spaces $H$ and $G$ as in [14] pp.20.

Also for two elements $\Phi,\Psi\in\boldsymbol{M}(\Lambda,{\mathscr{H}})$ the
operator cross covariance function $\Gamma_{\Phi,\Psi}$ will be defined as

(3.3)
$\Gamma_{\Phi,\Psi}(\lambda,\mu):=\left[\Phi(\lambda),\Psi(\mu)\right]_{{\mathscr{H}}};\quad\lambda,\mu\in\Lambda,$

while the scalar cross covariance function $\gamma_{\Phi,\Psi}$ is defined by

(3.4)
$\gamma_{\Phi,\Psi}(\lambda,\mu):=\mathrm{tr}\Gamma_{\Phi,\Psi}(\lambda,\mu);\quad\lambda,\mu\in\Lambda.$

When $\Phi=\Psi$, then we denote simply $\gamma_{\Phi,\Phi}=\ :\gamma_{\Phi}$
and $\Gamma_{\Phi,\Phi}=\ :\Gamma_{\Phi}$, which will be called the _scalar
covariance function_ , respectively the _operator covariance function_ of
$\Phi$.

Now, the operator covariance function $\Gamma_{\Phi}$ (the scalar one
$\gamma_{\Phi}$, respectively) of $\Phi$ is a $\mathscr{C}_{1}(H)$-valued
($\mathbb{C}$-valued) positive definite kernel on $\Lambda$, in the sense of
the positivity from $\mathcal{B}(H)$, i.e. it holds

(3.5)
$\sum\limits_{i,j\in\mathbb{N}_{m}}a_{i}\Gamma_{\Phi}(\lambda_{i},\lambda_{j})a_{j}^{\ast}\geq
0,$

for any $m\in\mathbb{N}$ and any finite systems
$a_{1},\ldots,a_{m}\in{\mathcal{B}}(H),\
\lambda_{1},\ldots,\lambda_{m}\in\Lambda$, (respectively in the sense of the
usual positivity in $\mathbb{C}$)

(3.6)
$\sum\limits_{i,j\in\mathbb{N}_{m}}\alpha_{i}\overline{\alpha}_{j}\gamma_{\Phi}(\lambda_{i},\lambda_{j})\geq
0$

for any $m\in\mathbb{N}$ and any finite systems
$\alpha_{1},\ldots,\alpha_{m}\in\mathbb{C},\lambda_{1},\ldots,\lambda_{m}\in\Lambda)$.

Indeed the relation (3.5) results by applying (3.3) and computing:

$\sum\limits_{i,j\in\mathbb{N}_{m}}a_{i}\Gamma_{\Phi}(\lambda_{i},\lambda_{j})a_{j}^{\ast}=\sum\limits_{i,j\in\mathbb{N}_{m}}a_{i}\left[\Phi(\lambda_{i}),\Phi(\lambda_{j})\right]_{{\mathscr{H}}}a_{j}^{\ast}=\\\
\sum\limits_{i,j\in\mathbb{N}_{m}}\left[a_{i}\Phi(\lambda_{i}),a_{j}\Phi(\lambda_{j})\right]_{{\mathscr{H}}}=\left[\sum\limits_{i\in\mathbb{N}_{m}}a_{i}\Phi(\lambda_{i}),\sum\limits_{j\in\mathbb{N}_{m}}a_{j}\Phi(\lambda_{j})\right]_{{\mathscr{H}}}\geqslant
0,$

while (3.6) appears also in [17].

Note also that the positive definite $\mathscr{C}_{1}(H)$ \- valued kernel
$\Gamma=\Gamma_{\Phi}$ reproduces a normal Hilbert ${\mathcal{B}}(H)$-module
${\mathscr{H}}_{\Gamma_{\Phi}}$ (see [14] Section 2.4, Thm. 13, pp.37), as
well as a Hilbert space $G_{\Gamma_{\Phi}}$ of $H$-valued functions on
$\Lambda$ (see [14], Prop. 23, Section 2.4, pp.44), while
$\gamma=\gamma_{\Phi}$ as complex valued positive definite kernel reproduces a
Hilbert space $K_{\gamma_{\Phi}}$, (see for example [2]).
It is not hard to infer now, extending the correspondence
$\Phi(\lambda)\mapsto\Gamma_{\Phi}(\lambda,\cdot)$ to a
${\mathcal{B}}(H)$-linear mapping between the generating submodules of
${\mathscr{H}}_{\Phi}$ and ${\mathscr{H}}_{\Gamma_{\Phi}}$, that these are
isomorph (i.e. ${\mathscr{H}}_{\Phi}\approxeq{\mathscr{H}}_{\Gamma_{\Phi}}$)
as normal Hilbert ${\mathcal{B}}(H)$-modules and also that the measurements
space $G_{\Phi}$ associated to $\Phi$ is isomorph to the Hilbert space
$G_{\Gamma_{\Phi}}$ ($G_{\Phi}\simeq G_{\Gamma_{\Phi}}$) having
$\Gamma_{\Phi}$ as operator reproducing kernel.

Moreover extending the correspondence
$\Phi(\lambda)\mapsto\gamma_{\Phi}(\lambda,\cdot)$ to a linear mapping between
generating subspaces of ${\mathscr{H}}_{(\Phi)}$ and of $K_{\gamma_{\Phi}}$,
respectively, we have that these are also isomorph as Hilbert spaces.

On the other hand, similarly to (3.2) the module isomorphisms

(3.7)
${\mathscr{H}}_{\Gamma_{\Phi}}\cong\mathscr{C}_{2}\left(G_{\Gamma_{\Phi}},H\right)\cong
H\widehat{\otimes}{G}_{\Gamma_{\Phi}}$

also hold.

Noticing that $G_{\Gamma_{\Phi}}$ is generated by elements of the form
$\Gamma_{\Phi}(\cdot,\lambda)x$ with $\lambda\in\Lambda,x\in H$, as well as
the fact that the normal Hilbert ${\mathcal{B}}(H)$-module isomorphism
${\mathscr{H}}_{\Phi}\approxeq{\mathscr{H}}_{\Gamma_{\Phi}}$ was constructed
by the ${\mathcal{B}}(H)$-linear extension of the correspondence
$\Phi(\lambda)\mapsto\Gamma_{\Phi}(\lambda,\cdot),$ the measurements space
$G_{\Phi}$ associated to the m.s.o. stochastic mapping $\Phi$ coincides with
the closed subspace in $L^{2}_{0}(\wp)$ generated by

(3.8) $\left\\{V^{*}_{\Phi(\lambda)}x\,,\,\lambda\in\Lambda,\ x\in
H\right\\},$

where $V$ is the module isomorphism (2.2).

The above results can be gathered in

######  Theorem 3.1.

Given an arbitrary set (a topological space) $\Lambda$ and a (continuous)
m.s.o. stochastic mapping $\Phi:\Lambda\rightarrow{\mathscr{H}}$ the following
assertions hold:

  1. (i)

the scalar covariance function $\gamma_{\Phi}$, the operator covariance
function $\Gamma_{\Phi}$ respectively, is a complex, a
$\mathscr{C}_{1}(H)$-valued respectively, positive definite (and continuous)
kernel on $\Lambda$;

  2. (ii)

the space ${\mathscr{H}}_{(\Phi)}$ (the module ${\mathscr{H}}_{\Phi}$
respectively), associated to $\Phi$ is isomorph as Hilbert space, (as normal
Hilbert ${\mathcal{B}}(H)$-module respectively) to the reproducing kernel
Hilbert space $K_{\gamma_{\Phi}}$, (the reproducing kernel normal Hilbert
${\mathcal{B}}(H)$-module ${\mathscr{H}}_{\Gamma_{\Phi}}$ respectively);

  3. (iii)

the measurements space $G_{\Phi}$, associated to $\Phi$ is isomorph to the
Hilbert space $G_{\Gamma_{\Phi}}$, having $\Gamma_{\Phi}$ as operatorial
reproducing kernel. In this way the two Hilbert spaces ensure respectively the
description of the ${\mathcal{B}}(H)$-modules
${\mathscr{H}}_{\Phi},{\mathscr{H}}_{\Gamma_{\Phi}}$ by means of the
operatorial models mentioned in (3.2), respectively (3.7).
Moreover, $G_{\Phi}$ can be described directly in terms of the m.s.o.
stochastic mapping $\Phi$ by (3.8).

The proof being contained in the considerations preceding the theorem we shall
only mention that, in the case where $\Lambda$ is a topological space, the
continuity of $\Phi$, easily implies the continuity of $\Gamma_{\Phi}$, which
in fact is equivalent to the continuity on the diagonal of
$\Lambda\times\Lambda$. ∎

The reproducing kernel technique for normal ${\mathcal{B}}(H)$-modules will be
also used to characterize the subordination of two m.s.o.s.m. in terms of
their operator (cross) covariance functions. Namely, in analogy to the
characterization given in [14] we have

######  Theorem 3.2.

If $\Phi$ and $\Psi$ are two m.s.o.s.m. having the operator covariance
functions $\Gamma_{\Phi}$ and $\Gamma_{\Psi}$ respectively, and the operator
cross covariance function $\Gamma_{\Phi,\Psi}$, then $\Phi$ is subordinate to
$\Psi$, iff the function
$K_{\lambda}(\cdot)=\Gamma_{\Phi,\Psi}(\lambda,\cdot)$ is in the
${\mathcal{B}}(H)$-module ${\mathscr{H}}_{\Gamma_{\Psi}}$ with $\Gamma_{\Psi}$
as reproducing kernel and

(3.9)
$[K_{\lambda},K_{\mu}]_{{\mathscr{H}}_{\Gamma_{\Psi}}}=\Gamma_{\Phi}(\lambda,\mu)\,;\quad\lambda,\,\mu\in\Lambda.$

The proof is based on the fact that the operator cross covariance function
$\Gamma_{\Phi,\Psi}$ of $\Phi,\Psi\in\boldsymbol{M}(\Lambda,{\mathscr{H}})$
appears in the expression of the operator covariance function of the product
$\Phi\times\Psi\in\boldsymbol{M}(\Lambda,{\mathscr{H}}\times{\mathscr{H}})$
(${\mathscr{H}}\times{\mathscr{H}}$ being the “product”
${\mathcal{B}}(H)$-module as in [14, Section 4.9, pp. 192]), namely

$\Gamma_{\Phi\times\Psi}(\lambda,\mu)=\begin{pmatrix}\Gamma_{\Phi}(\lambda,\mu)&\Gamma_{\Phi,\Psi}(\lambda,\mu)\\\
\Gamma_{\Psi,\Phi}(\lambda,\mu)&\Gamma_{\Psi}(\lambda,\mu)\end{pmatrix},\quad\lambda,\mu\in\Lambda.$

Having in view the preceding results, the rest of the proof runs similarly as
in Theorem 9.3, pp. 193 from [14, Chap.4]. ∎

If we consider the operator correlation mapping

(3.10)
$\mathbf{M}(\Lambda,{\mathscr{H}})\ni\Phi\mapsto\Gamma_{\Phi}\in\mathbf{\Gamma}\left(\Lambda,\mathscr{C}_{1}(H)\right),$

from the set of m.s.o.s.m. $\mathbf{M}(\Lambda,{\mathscr{H}})$ to the set of
all positive definite ${\mathscr{C}}_{1}(H)$-valued kernels on $\Lambda$ (see
(3.5)), $\mathbf{\Gamma}\left(\Lambda,\mathscr{C}_{1}(H)\right)$, then the
next theorem proves its surjectivity and gives a description of the pre-image
of each $\Gamma\in\mathbf{\Gamma}\left(\Lambda,\mathscr{C}_{1}(H)\right)$.

######  Theorem 3.3.

  1. (i)

If $\Lambda$ is an arbitrary set (a topological space), then for any positive
definite $\mathscr{C}_{1}(H)$ \- valued kernel $\Gamma$ on $\Lambda$
(continuous on $\Lambda\times\Lambda$), there exists a (continuous)
multivariate second order stochastic mapping
$\Phi:\Lambda\rightarrow{\mathscr{H}}$, which assures for $\Gamma$ a
Kolmogorov factorization:

(3.11) $\Gamma(\lambda,\mu)=[\Phi(\lambda),\Phi(\mu)]_{{\mathscr{H}}}\ ;\
\lambda,\mu\in\Lambda,$

i.e. $\Gamma$ coincides with the operator covariance function $\Gamma_{\Phi}$
of $\Phi$.

  2. (ii)

If $\Phi_{1}$ and $\Phi_{2}$ are two (continuous) m.s.o. stochastic mappings
on $\Lambda$ for which $\Gamma_{\Phi_{1}}=\Gamma_{\Phi_{2}}$, then $\Phi_{1}$
and $\Phi_{2}$ are gramian unitary equivalent, i.e. there exists a gramian
unitary operator
$\mathrm{W}:{\mathscr{H}}_{\Phi_{2}}\rightarrow{\mathscr{H}}_{\Phi_{1}}$ such
that

(3.12)
$\Phi_{1}(\lambda)=\mathrm{W}\Phi_{2}(\lambda)\,,\quad\lambda\in\Lambda.$

###### Proof.

(i) Let $\Gamma$ be a positive definite ${\mathscr{C}}_{1}(H)$-valued kernel
on $\Lambda$ and ${\mathscr{H}}_{\Gamma}$ ($G_{\Gamma}$ respectively) the
normal Hilbert ${\mathcal{B}}(H)$-module (Hilbert space respectively)
reproduced by $\Gamma$, generated as in [14] Sec.2.4,Thm.13,pp.37 or
Prop.23,pp.44, with the gramian (respectively the scalar product) defined
there. Since, by Thm. 2.4., pp. 45 of [14], ${\mathscr{C}}_{2}(G_{\Gamma},H)$
is an operatorial model for ${\mathscr{H}}_{\Gamma}$, embedding $G_{\Gamma}$
into a space of the form $L^{2}_{0}(\wp)$ (see [18]) ${\mathscr{H}}_{\Gamma}$
embeds into a normal Hilbert ${\mathcal{B}}(H)$-module of the form
$\mathscr{C}_{2}\left(L^{2}_{0}(\wp),H\right)$.
In this way the ${\mathscr{H}}_{\Gamma}$-valued function defined by
$\Phi(\lambda):=\Gamma(\lambda,\cdot),\ (\lambda\in\Lambda)$ gives us exactly
the m.s.o. stochastic mapping we seek, since we get for any
$\lambda,\mu\in\Lambda$

$\Gamma_{\Phi}(\lambda,\mu)=\left[\Phi(\lambda),\Phi(\mu)\right]_{{\mathscr{H}}}=\left[\Gamma(\lambda,\cdot),\Gamma(\mu,\cdot)\right]_{{\mathscr{H}}_{\Gamma}}=\Gamma(\lambda,\mu).$

(ii) From the hypothesis on $\Phi_{1}$ and $\Phi_{2}$ we have

$\left[\Phi_{1}(\lambda),\Phi_{1}(\mu)\right]_{{\mathscr{H}}}=\left[\Phi_{2}(\lambda),\Phi_{2}(\mu)\right]_{{\mathscr{H}}},\quad\lambda,\mu\in\Lambda,$

which means that the mapping $\mathrm{W}$ defined on
$\left\\{\Phi_{2}(\mu),\mu\in\Lambda\right\\}$ by

$\mathrm{W}\Phi_{2}(\mu)=\Phi_{1}(\mu)\,,\quad\mu\in\Lambda$

preserves the gramian, is easily extended by ${\mathcal{B}}(H)$-linearity,
still conserving the gramian and is clearly surjective between the
${\mathcal{B}}(H)$-modules ${\mathscr{H}}_{\Phi_{2}}^{0}$ and
${\mathscr{H}}_{\Phi_{1}}^{0}$ (see (3.1)). This allows the extension of
${\mathrm{W}}$ by continuity to a gramian unitary operator from
${\mathscr{H}}_{\Phi_{2}}$ to ${\mathscr{H}}_{\Phi_{1}}$, satisfying (3.12). ∎

######  Remark 3.1.

Analogous results, formulated in a Hilbert space setting, hold for the scalar
correlation mapping

(3.13)
$\boldsymbol{M}(\Lambda,{\mathscr{H}})\ni\Phi\mapsto\gamma_{\Phi}\in\boldsymbol{\gamma}(\Lambda),$

where $\boldsymbol{\gamma}(\Lambda)$ is the set of complex valued positive
definite kernels on $\Lambda$.

##  4\. Multivariate second order random distribution fields and their
covariance distributions

In this Section we illustrate how the results from the previous Section can be
applied to the case of multivariate second order (m.s.o.) random distribution
fields.

For completeness we shall work with the _$m$ -order $(0\leqslant
m\leqslant\infty)$ m.s.o. random distribution field_ $U$, which is a
continuous linear m.s.o stochastic mapping having the index set
$\Lambda={\mathcal{D}}_{d}^{m}$, the space of complex valued compactly
supported continuously derivable functions up to the order
$m\,(m=0,1,2,\ldots,\infty)$. In other words
$U\in\big{(}{\mathcal{D}}^{m}_{d}\big{)}^{\prime}({\mathscr{H}}):={\mathcal{B}}({\mathcal{D}}_{d}^{m},{\mathscr{H}})$,
i.e. it is a ${\mathscr{H}}$-valued $m$-order distribution on $\mathbb{R}^{d}$
(see also [21]). For $m=0$, $U$ will be a (${\mathscr{H}}$-valued) m.s.o.
random Radon measure and for $m=\infty$, the index $m$ and the “$m$-order”
will be omitted, otherwise $U$ will be called of _finite $m$-order_.

Now for $U,V\in\big{(}{\mathcal{D}}^{m}_{d}\big{)}^{\prime}({\mathscr{H}})$
the scalar cross covariance function $\gamma_{U,V}$ and the operator cross
covariance function $\Gamma_{U,V}$ are defined by (3.4) and (3.3). We shall
comment only the properties of $\Gamma_{U,V}$, which – from the linearity and
continuity of $U$ and $V$ – is sesquilinear and continuous on
${\mathcal{D}}_{d}^{m}\times{\mathcal{D}}_{d}^{m}$. Moreover $\Gamma_{U}$ is
obviously positive definite, i.e. it satisfies a relation of the form (3.5).

It is possible as in Theorems 3.1 and 3.3 to describe and to characterize the
m.s.o. random distribution fields, their modular and vector domains and
measurements space in terms of such kernels. We prefer but to do that by using
everywhere a distributional framework, where instead of
${\mathscr{C}}_{1}(H)$-valued sesquilinear continuous kernels on
${\mathcal{D}}_{d}^{m}\times{\mathcal{D}}_{d}^{m}$, we use the so-called
distribution kernels on ${\mathbb{R}}^{d}$ in the sense of L. Schwartz (see
[20, I, pp. 138]), which are $m$ \- order distributions on $\mathbb{R}^{2d}$.
Therefore we need the following Lemma, where we identify
${\mathcal{D}}_{2d}^{m}$ with the inductive tensor product
${\mathcal{D}}_{d}^{m}\otimes_{i}{\mathcal{D}}_{d}^{m}$ (see for example [10,
pp. 84]).

######  Lemma 4.1.

Let $\Gamma$ be a ${\mathscr{C}}_{1}(H)$-valued sesquilinear kernel on
${\mathcal{D}}_{d}^{m}$, which is continuous on the diagonal of
${\mathcal{D}}_{d}^{m}\times{\mathcal{D}}_{d}^{m}$. Then there is an $m$-order
distribution kernel $C_{\Gamma}$ on ${\mathbb{R}}^{d}$ such that

(4.1)
$C_{\Gamma}(\varphi\otimes\psi)=\Gamma(\varphi,\bar{\psi}),\quad\varphi,\psi\in{\mathcal{D}}_{d}^{m}.$

Moreover, when $\Gamma$ is positive on the diagonal, then $C_{\Gamma}$ is a
positive definite $m$-order distribution kernel.

###### Proof.

First, from the hypothesis, it is easily seen that the kernel $\Gamma$ is
continuous on ${\mathcal{D}}_{d}^{m}\times{\mathcal{D}}_{d}^{m}$ and when
$\Gamma(\varphi,\varphi)\geq 0,\ \varphi\in{\mathcal{D}}_{d}^{m}$, then
$\Gamma$ is also positive definite. We shall attach an $m$-order
$\mathscr{C}_{1}(H)$ \- valued distribution $C_{\Gamma}$ on $\mathbb{R}^{2d}$
as follows. Define first $C_{\Gamma}$ on the elementary tensors
$\varphi\otimes\psi$ from ${\mathcal{D}}^{m}_{d}\otimes{\mathcal{D}}^{m}_{d}$
by $C_{\Gamma}(\varphi\otimes\psi):=\Gamma(\varphi,\overline{\psi})$, extend
that by linearity, and then by continuity to the whole
${\mathcal{D}}^{m}_{d}\widehat{\otimes}_{i}{\mathcal{D}}^{m}_{d}={\mathcal{D}}^{m}_{2d}$
preserving the notation $C_{\Gamma}$.
So
$C_{\Gamma}\in\big{(}{\mathcal{D}}^{m}_{2d}\big{)}^{\prime}(\mathscr{C}_{1}(H))$,
which, when $\Gamma$ is positive on the diagonal, will be a positive definite
distribution kernel on ${\mathbb{R}}^{d}$. We shall refer to this as
$C_{\Gamma}\in
pd({\mathcal{D}}^{m}_{2d})^{\prime}\big{(}{\mathscr{C}}_{1}(H)\big{)}$. ∎

Since, for each
$U,V\in\big{(}{\mathcal{D}}^{m}_{d}\big{)}^{\prime}({\mathscr{H}})$ the
operator cross covariance functions $\Gamma_{U,V}$, $\Gamma_{U}$ satisfy the
hypotheses from Lemma 4.1, the existence of the distribution kernels
$C_{\Gamma_{U,V}}$ and $C_{\Gamma_{U}}$ is assured, the last one being even
positive definite. These will be called the _operator cross covariance
distribution_ of $U$ and $V$, respectively the _operator covariance
distribution of_ the m.s.o. random distribution field _$U$_ and denoted by
$C_{U,V}$, respectively $C_{U}$. Correspondingly $c_{U,V}$ defined by
$c_{U,V}(\chi):=trC_{U,V}(\chi),\,\chi\in{\mathcal{D}}^{m}_{2d}$, will be
called the _scalar cross covariance distribution_ of $U$ and $V$, respectively
$c_{U}=c_{U,U}$ the _scalar covariance distribution_ of $U$.
By applying Theorem 3.1, the following description of the domains associated
to a m.s.o.r.d.f. in terms of its covariance distribution holds.

######  Corollary 4.2.

For a given m.s.o. random distribution field
$U\in\big{(}{\mathcal{D}}^{m}_{d}\big{)}^{\prime}({\mathscr{H}})$, the modular
domain ${\mathscr{H}}_{U}$, respectively the measurements space $G_{U}$ will
be isomorph to the ${\mathcal{B}}(H)$-module ${\mathscr{H}}_{\Gamma^{C_{U}}}$,
respectively the Hilbert space $G_{\Gamma^{C_{U}}}$ both reproduced by the
$\mathscr{C}_{1}(H)$-valued kernel $\Gamma^{C_{U}}$, while the vector domain
${\mathscr{H}}_{(U)}$ of $U$ is isomorphic to the Hilbert space reproduced by
the complex valued kernel $\gamma^{C_{U}}$.

The operator cross covariance distribution together with the operator
covariance distributions of two m.s.o.r.d.f. will be used to obtain from
Theorem 3.2 the following characterization of their subordination.

######  Corollary 4.3.

Given two m.s.o.r.d.f. $U$ and $V$ then $U$ is subordinate to $V$, iff for
each fixed $\varphi\in{\mathcal{D}}_{d}$, the function $K_{\varphi}(\cdot)$
given by

${\mathcal{D}}_{d}\ni\psi\mapsto
C_{U,V}(\varphi\otimes\bar{\psi})\in{\mathscr{C}}_{1}(H)$

lies in the ${\mathcal{B}}(H)$-module ${\mathscr{H}}_{\Gamma^{C_{V}}}$ and
$\bigl{[}K_{\varphi},K_{\psi}\bigr{]}_{{\mathscr{H}}_{\Gamma^{C_{V}}}}=C_{U}(\varphi\otimes\bar{\psi})$.

The operator correlation mapping (3.10) becomes

(4.2) $({\mathcal{D}}_{d}^{m})^{\prime}({\mathscr{H}})\ni U\mapsto C_{U}\in
pd({\mathcal{D}}_{2d}^{m})^{\prime}({\mathscr{C}}_{1}(H)),$

which we shall call the _operator covariance distribution mapping_. Its
properties are contained in the following Corollaries.

######  Corollary 4.4.

For an $m$-order distribution kernel on ${\mathbb{R}}^{d}$,
$C\in\big{(}{\mathcal{D}}^{m}_{2d}\big{)}^{\prime}(\mathscr{C}_{1}(H))$ the
following statements are equivalent:

  * (i)

$C\in pd\big{(}{\mathcal{D}}^{m}_{2d}\big{)}^{\prime}(\mathscr{C}_{1}(H))$;
i.e. the operatorial kernel $\Gamma^{C}$ on ${\mathcal{D}}^{m}_{d}$ defined by
$\Gamma^{C}(\varphi,\psi):=C(\varphi\otimes\overline{\psi})\
(\varphi,\psi\in{\mathcal{D}}^{m}_{d})$ is positive definite.

  * (ii)

The operatorial kernel $\Gamma^{C}$ is sesquilinear and positive (in the sense
of positivity from ${\mathcal{B}}(H)$) on the diagonal of
${\mathcal{D}}^{m}_{d}\times{\mathcal{D}}^{m}_{d}$.

  * (iii)

There is a m.s.o. random distribution field
$U\in\big{(}{\mathcal{D}}^{m}_{d}\big{)}^{\prime}(\mathscr{H})$ such that
$C=C_{U}$ (i.e. $C$ has a Kolmogorov type factorization).

The uniqueness (up to a gramian unitary equivalence) of $U$ in (iii) of the
previous Corollary results now from Theorem 3.3(ii). More precisely it holds

######  Corollary 4.5.

If the m.s.o. random distribution fields
$U^{1},\,U^{2}\in\big{(}{\mathcal{D}}^{m}_{d}\big{)}^{\prime}({\mathscr{H}})$
have the same operator covariance distribution, i.e. $C_{U^{1}}=C_{U^{2}}$,
then there exists a gramian unitary operator
$W:{\mathscr{H}}_{U^{1}}\rightarrow{\mathscr{H}}_{U^{2}}$ such that

(4.3)
$W\,U^{1}_{\varphi}=U^{2}_{\varphi},\quad\varphi\in{\mathcal{D}}^{m}_{d}.$

It would be of course interesting to know _if each
${\mathscr{C}}_{1}(H)$-valued distribution kernel on ${\mathbb{R}}^{d}$ is the
operator cross covariance distribution of some pair
$U,V\in({\mathcal{D}}_{d}^{m})^{\prime}({\mathscr{H}})$_, but this won’t be
our goal here.
Now we shall introduce in our general frame the concepts of determinism and
nondeterminism and we give a general decomposition of Wold type of a
m.s.o.r.d.f., $U$, into deterministic and purely nondeterministic parts, both
summands being operator subordinate to $U$.

In [19] (see also [1]), the observable space up to the moment $t_{0}$ for a
random distribution for the case $d=1$ and $H=\mathbb{C}$ was defined. By
using in ${\mathbb{R}}^{d}$ the ordering relation $s=(s_{1},\dots,s_{d})\leq
t=(t_{1},\dots,t_{d})$ if $s_{j}\leq t_{j},\ j=1,\dots,d$, we shall define by
analogy these observable structures for m.s.o.r.d.f. up to the moment $t_{0}$.
To this purpose we introduce first the subspace ${\mathcal{D}}_{d}^{t_{0}}$ in
${\mathcal{D}}_{d}$ by

(4.4)
$\varphi\in{\mathcal{D}}_{d}^{t_{0}}\quad\Leftrightarrow\quad\textrm{supp}\varphi\subset\left\\{t\in\mathbb{R}^{d}\,:\,t\leqslant
t_{0}\right\\}.$

Thus, for a m.s.o.r.d.f.
$U=\\{U_{\varphi}\\}_{\varphi\in{\mathcal{D}}_{d}^{m}}$ we call the observable
module ${\mathscr{H}}_{U}^{t_{0}}$ (the observable space
${\mathscr{H}}_{(U)}^{t_{0}}$) up to the moment $t_{0}$, the closed
${\mathcal{B}}(H)$-module (space) in ${\mathscr{H}}$, generated by the set
$\left\\{U_{\varphi}\quad,\,\varphi\in{\mathcal{D}}_{d}^{t_{0}}\right\\}$.

We notice that the Hilbert ${\mathcal{B}}(H)$-module (Hilbert space) generated
by $\bigcup\limits_{t\in\mathbb{R}^{d}}{\mathscr{H}}_{U}^{t}$
($\bigcup\limits_{t\in\mathbb{R}^{d}}{\mathscr{H}}_{(U)}^{t}$ respectively) is
just the modular domain ${\mathscr{H}}_{U}$ (vector domain respectively) of
the m.s.o.r.d.f. $U$, while
$\bigcap\limits_{t\in\mathbb{R}^{d}}{\mathscr{H}}_{U}^{t}$
($\bigcap\limits_{t\in\mathbb{R}^{d}}{\mathscr{H}}_{(U)}^{t}$ respectively)
will be called the _remote past_ module (space) of $U$ and denoted by
${\mathscr{H}}_{U}^{-\infty}$ (${\mathscr{H}}_{(U)}^{-\infty}$ respectively).
We shall say that $\varphi\mapsto U_{\varphi}$ is operator deterministic (or,
simply deterministic), if ${\mathscr{H}}_{U}={\mathscr{H}}_{U}^{-\infty}$
(${\mathscr{H}}_{(U)}={\mathscr{H}}_{(U)}^{-\infty}$ respectively). If
${\mathscr{H}}_{U}^{-\infty}\subsetneqq{\mathscr{H}}_{U}$
(${\mathscr{H}}_{(U)}^{-\infty}\subsetneqq{\mathscr{H}}_{(U)}$ respectively)
we say that $U$ is operator nondeterministic (nondeterministic).

For extreme situations, where ${\mathscr{H}}_{U}^{-\infty}=\\{0\\}$ ,
(${\mathscr{H}}_{(U)}^{-\infty}=\\{0\\}$ respectively), the term _non-
deterministic_ becomes _purely non-deterministic_ (see also [32], [14, Sec.
5.1] for the first, respectively the classical result). In this context we
have the following general Wold decomposition for m.s.o.r.d.f.

######  Theorem 4.6.

For a m.s.o.r.d.f. $\\{U_{\varphi}\\}_{\varphi\in{\mathcal{D}}_{d}}$ taking
values in ${\mathscr{H}}$, there exists a unique decomposition

(4.5)
$U_{\varphi}=U_{\varphi}^{det}+U_{\varphi}^{p}\,,\quad\varphi\in{\mathcal{D}}_{d}$

of $U$ such that

  1. (i)

$\\{U_{\varphi}^{det}\\}_{\varphi\in{\mathcal{D}}_{d}}$ is operator
deterministic and $\\{U_{\varphi}^{p}\\}_{\varphi\in{\mathcal{D}}_{d}}$ is
operator purely nondeterministic,

  2. (ii)

$U^{det}$ and $U^{p}$ are operator subordinate to $U$,

  3. (iii)

$U^{det}$ and $U^{p}$ are operator uncorrelated, i.e. the operator cross
covariance distribution $C_{U^{det},U^{p}}$ vanishes on ${\mathcal{D}}_{2d}$.

Moreover the gramian orthogonal decomposition

${\mathscr{H}}_{U}={\mathscr{H}}_{U^{det}}\oplus{\mathscr{H}}_{U^{p}}$

holds.

###### Proof.

Since in normal Hilbert ${\mathcal{B}}(H)$-modules, any closed submodule has a
gramian projection (Lemma 2, Section 2.2, pp.22 [14]), let $P$ be the gramian
projection associated to the submodule ${\mathscr{H}}_{U}^{-\infty}$ and let

(4.6) $U_{\varphi}^{det}=PU_{\varphi}\quad,\quad
U_{\varphi}^{p}=(I-P)U_{\varphi}\,,\quad\varphi\in{\mathcal{D}}_{d}.$

One can verify by using standard arguments (as in [14, Section 5.1]) that
(4.6) gives us the very m.s.o.r.d.f. we seek, the uniqueness of decomposition
(4.5) being obtained by direct reasoning. ∎

##  5\. Multivariate second order stochastic measures and their associated
bimeasures

In this Section we shall see how from the theory of m.s.o.r.d.f. we can indeed
infer the classical theory of m.s.o. random fields, as well as how to fit in
this general theory the special class of m.s.o.r.d.f., namely that of (not
necessarily bounded) measures, which is very useful for the spectral
representations of random (distribution) fields.
Each very well known continuous m.s.o. random field $F$ on $\mathbb{R}^{d}$
(i.e. $F\in\mathcal{E}^{0}_{d}({\mathscr{H}})$) will be identified with the
m.s.o. random Radon measure $U^{F}$ (i.e. m.s.o. random distribution fields of
zero order having as index set ${\mathcal{D}}_{d}^{0}$), defined by

(5.1)
$U_{\varphi}^{F}=\int\limits_{\mathbb{R}^{d}}\varphi(t)F(t)dt,\quad\,\varphi\in{\mathcal{D}}^{0}_{d}.$

Moreover, since $F$ is bounded on each bounded Borel subset $A$ of
${\mathbb{R}}^{d}$ (i.e. on each
$A\in\widetilde{\mathcal{B}}or({\mathbb{R}}^{d})$), it can be even regarded as
a not necessarily bounded regular measure $\xi^{F}$ through

(5.2) $\xi^{F}(A)=\int\limits_{A}F(t)dt,\quad
A\in\widetilde{\mathcal{B}}or(\mathbb{R}^{d}).$

In such a way the locally convex ${\mathcal{B}}(H)$-module
${\mathcal{E}}_{d}^{0}({\mathscr{H}})$ is embedded into the class of
_multivariate second order stochastic, not necessarily bounded, regular
measures_ , for which the index set $\Lambda$ is the $\delta$-ring
$\widetilde{{\mathcal{B}}}or(\mathbb{R}^{d})$. Playing an important role in
the definitions of various kinds of harmonizability of m.s.o. random
distribution fields, they are often supposed to have finite operator
semivariation i.e.

(5.3) $\|\xi\|_{o}(A)<\infty,\quad
A\in\widetilde{{\mathcal{B}}}or(\mathbb{R}^{d}),$

where $\|\cdot\|_{o}$ is defined as in [14, pp. 56 Def.4 (1)], or finite
semivariation i.e.

(5.4) $\|\xi\|(A)<\infty,\quad
A\in\widetilde{{\mathcal{B}}}or(\mathbb{R}^{d}),$

where $\|\cdot\|$ is defined as in [14, pp.54 Def.1], the corresponding
classes of m.s.o. stochastic regular measures being denoted by
$fosvr\mathcal{M}_{d}({\mathscr{H}})$, or by
$fsvr\mathcal{M}_{d}({\mathscr{H}})$, respectively. They are locally convex
${\mathcal{B}}(H)$-modules with the seminorms given by (5.3) and (5.4)
respectively. However, since
$\|\xi\|(A)\leqslant\|\xi\|_{0}(A),\,A\in\widetilde{{\mathcal{B}}}or(\mathbb{R}^{d})$,
the inclusion $fosvr\mathcal{M}_{d}({\mathscr{H}})\subset\linebreak
fsvr\mathcal{M}_{d}({\mathscr{H}})$ holds with continuous embedding.

Now an element $\xi\in fsvr\,{\mathcal{M}}_{d}({\mathscr{H}})$ is to be
regarded as a ${\mathscr{H}}$-valued distribution $U^{\xi}$, which will be
even a ${\mathscr{H}}$ \- valued Radon measure, by putting

(5.5)
$U_{\varphi}^{\xi}=\int\limits_{\mathbb{R}^{d}}\varphi(t)d\xi(t),\quad\varphi\in{\mathcal{D}}_{d}^{0}.$

Thus, we have obtained

######  Proposition 5.1.

For the above mentioned classes of m.s.o. random distribution fields the
following inclusions hold with continuous embedding

(5.6) ${\mathcal{E}}^{0}_{d}({\mathscr{H}})\subset
fosvr\,{\mathcal{M}}_{d}({\mathscr{H}})\subset
fsvr\,{\mathcal{M}}_{d}({\mathscr{H}})\subset\\\
\subset({\mathcal{D}}_{d}^{0})^{\prime}({\mathscr{H}})\subset({\mathcal{D}}_{d}^{m})^{\prime}({\mathscr{H}})\subset{\mathcal{D}}^{\prime}_{d}({\mathscr{H}}).$

Let’s observe that a m.s.o. stochastic mapping from one of the mentioned
classes can have more than one index set. For example, looking to the above
identifications:

$\xi=\\{\xi(A)\\}_{A\in\widetilde{\mathscr{B}}or({\mathbb{R}}^{d})}\ \text{
with }\ U^{\xi}=\\{U^{\xi}_{\varphi}\\}_{\varphi\in{\mathcal{D}}_{d}^{0}},$

respectively

$F=\\{F(t)\\}_{t\in{\mathbb{R}}^{d}}\ \text{ with }\
\xi^{F}=\\{\xi^{F}(A)\\}_{A\in\widetilde{\mathscr{B}}or({\mathbb{R}}^{d})}\
\text{ and }\ U^{F}=U^{\xi^{F}}$

it is not difficult to see that

${\mathscr{H}}_{\xi}={\mathscr{H}}_{U^{\xi}},\ \text{ respectively }\
{\mathscr{H}}_{F}={\mathscr{H}}_{\xi^{F}}={\mathscr{H}}_{U^{F}},$

analogous relations being true for the corresponding vector domains and
measurements spaces. It is of interest how those classes of positive definite
kernels which are co-domains of the “restrictions” of the operator covariance
distribution mapping (4.2) to the submodules from (5.6) can be suitable
described. Let’s mention that for the operator covariance distribution of an
element $\xi\in fsvr{\mathcal{M}}_{d}({\mathscr{H}})$ regarded as a m.s.o.
random distribution field we naturally use the notation $\Gamma_{U^{\xi}}$,
while if $\xi$ is regarded as a m.s.o. stochastic mapping on
$\widetilde{\mathcal{B}}or({\mathbb{R}}^{d})$, then its operator covariance
function $\Gamma_{\xi}$ represents a positive definite regular bimeasure on
$\widetilde{\mathcal{B}}or({\mathbb{R}}^{d})\times\widetilde{\mathcal{B}}or({\mathbb{R}}^{d})$,
for which we shall use also the notation $\tau_{\xi}$. It will be also called
the _operator covariance bimeasure_ associated to the m.s.o stochastic measure
$\xi$.

For the bimeasures on
$\widetilde{\mathcal{B}}or({\mathbb{R}}^{d})\times\widetilde{\mathcal{B}}or({\mathbb{R}}^{d})$
and their semivariation or operator semivariation we adopt analogue
definitions as in [14] (Definition 9 (1) and (3) pp.62 and Definition 16
pp.65). For the spaces of such (${\mathscr{C}}_{1}(H)$-valued) regular
bimeasures with finite (operator) semivariation we shall use the notation
$fsvr\,\mathfrak{M}_{2d}\big{(}{\mathscr{C}}_{1}(H)\big{)}$
($fosvr\,\mathfrak{M}_{2d}\big{(}{\mathscr{C}}_{1}(H)\big{)}$, respectively).
However, since in our case, the measure and the bimeasure are defined on a
$\delta$-ring, some properties from [14] do not automatically hold.

The corresponding classes of positive definite bimeasures will be denoted by
$fsvr\,\mathfrak{M}_{2d}^{pd}\big{(}{\mathscr{C}}_{1}(H)\big{)}$ and
$fosvr\,\mathfrak{M}_{2d}^{pd}\big{(}{\mathscr{C}}_{1}(H)\big{)}$. So in the
above notation $\tau_{\xi}\in
f(o)svr{\mathfrak{M}}_{2d}^{pd}({\mathscr{C}}_{1}(H))$.
Let’s also mention that it is not hard to see that a Morse-Transue (MT)_strict
integral_ as in [14, Section 1.2 pp.5] can be defined for bimeasures $\tau$ on
$\widetilde{\mathcal{B}}or({\mathbb{R}}^{d})\times\widetilde{\mathcal{B}}or({\mathbb{R}}^{d})$.
With such a strict MT-integral to each $\tau\in
fsvr{\mathfrak{M}}_{2d}({\mathscr{C}}_{1}(H))$ we can attach a distribution
$C^{\tau}$ on ${\mathbb{R}}^{2d}$ (a distribution kernel on
${\mathbb{R}}^{d}$, in the sense of L. Schwartz), first defined on elementary
tensors trough

(5.7)
$C^{\tau}(\varphi\otimes\psi)=\int\limits_{\mathbb{R}^{d}}\int\limits_{\mathbb{R}^{d}}\varphi(s)\psi(t)d\tau(s,t),\quad\varphi,\psi\in{\mathcal{D}}_{d}^{m},$

and then, by the usual extension, to the whole ${\mathcal{D}}^{m}_{2d}$.

On the other hand, each
$K\in{\mathcal{E}}^{0}_{2d}\big{(}{\mathscr{C}}_{1}(H)\big{)}$ can be regarded
as a regular bimeasure $\tau^{K}$, having finite operator semivariation, by
putting

(5.8) $\tau^{K}(A,B):=\int\limits_{A}\int\limits_{B}K(s,t)dsdt,\quad
A,B\in\widetilde{{\mathcal{B}}}or(\mathbb{R}^{d}).$

This infers

######  Proposition 5.2.

The valued domains of the restrictions of the operator covariance distribution
mapping (4.2) to the spaces from (5.6) satisfy respectively the following
inclusions

(5.9) $pd{\mathcal{E}}^{0}_{2d}\big{(}{\mathscr{C}}_{1}(H)\big{)}\subset
fosvr\,\mathfrak{M}_{2d}^{pd}\big{(}{\mathscr{C}}_{1}(H)\big{)}\subset
fsvr\,\mathfrak{M}_{2d}^{pd}\big{(}{\mathscr{C}}_{1}(H)\big{)}\subset\\\
\subset
pd({\mathcal{D}}^{0}_{2d})^{\prime}\big{(}{\mathscr{C}}_{1}(H)\big{)}\subset
pd({\mathcal{D}}^{m}_{2d})^{\prime}\big{(}{\mathscr{C}}_{1}(H)\big{)}\subset
pd({\mathcal{D}}_{2d})^{\prime}\big{(}{\mathscr{C}}_{1}(H)\big{)}.$

For a more complete image we discuss in detail the restrictions of the
operator covariance distribution mapping (4.2) to the space
$fsvr\,{\mathcal{M}}_{d}({\mathscr{H}})$, respectively to
$fosvr\,{\mathcal{M}}_{d}({\mathscr{H}})$ and more particular to
${\mathcal{E}}^{0}_{d}({\mathscr{H}})$. Since

$\tau_{\xi}(A,B)=(\xi\otimes\xi)(A,B)=[\xi(A),\xi(B)]_{{\mathscr{H}}},\quad
A,B\in\widetilde{{\mathcal{B}}}or(\mathbb{R}^{d})$

we can apply Lemma 19(1) and (3) of [14, pp. 66], from where we deduce that
$\tau_{\xi}\in
fsvr\,\mathfrak{M}_{2d}^{pd}\big{(}{\mathscr{C}}_{1}(H)\big{)}$, respectively
$\tau_{\xi}\in
fosvr\,\mathfrak{M}_{2d}^{pd}\big{(}{\mathscr{C}}_{1}(H)\big{)}$.

It is not difficult to see that it results

######  Proposition 5.3.

Statements analogous as for the operator covariance distribution mapping (4.2)
in the corollaries above, hold for the corresponding mappings

(5.10)
$\begin{split}fosvr\,{\mathcal{M}}_{d}({\mathscr{H}})\ni\xi&\mapsto\tau_{\xi}\in
fosvr\,\mathfrak{M}_{2d}^{pd}\big{(}{\mathscr{C}}_{1}(H)\big{)}\\\
fsvr\,{\mathcal{M}}_{d}({\mathscr{H}})\ni\xi&\mapsto\tau_{\xi}\in
fsvr\,\mathfrak{M}_{2d}^{pd}\big{(}{\mathscr{C}}_{1}(H)\big{)}.\end{split}$

Moreover, these operator covariance bimeasure mappings are natural extensions
of the covariance function mapping

(5.11) ${\mathcal{E}}^{0}_{d}({\mathscr{H}})\ni F\mapsto\Gamma_{F}\in
pd{\mathcal{E}}^{0}_{2d}\big{(}{\mathscr{C}}_{1}(H)\big{)},$

associated to classical m.s.o. random fields, as was defined in [14, Section
4.1, pp. 148].

Now it is naturally to ask how the mappings (5.11),(5.10) can be regarded as
restrictions of the covariance distribution mapping (4.2) to the first three
subspaces from (5.6), i.e our generalization is coherent to the classical case
in [14].
We shall show that the operator covariance distributions of (5.10) and of
(5.11) will be regarded as ${\mathscr{C}}_{1}(H)$-valued distribution
$C^{\tau_{\xi}}$ on ${\mathbb{R}}^{2d}$ corresponding to (generated by) the
bimeasure $\tau_{\xi}$, respectively $C^{\Gamma_{F}}$ corresponding to
(generated by) the correlation function $\Gamma_{F}$. More precisely it holds

######  Proposition 5.4.

The positive definite operator valued bimeasures from (5.10) and the operator
covariance function of (5.11) satisfy

(5.12) $C_{U^{\xi}}=C^{\tau_{\xi}}\ \text{ and }\ C_{U^{F}}=C^{\Gamma_{F}},$

respectively.

###### Proof.

Let $\xi\in fosvr\,{\mathcal{M}}_{d}({\mathscr{H}})$, and
$U^{\xi}\in({\mathcal{D}}_{d}^{m})^{\prime}({\mathscr{H}})$ given by (5.5).
Then we successively obtain for each
$\varphi,\psi\in{\mathcal{D}}_{d}^{m}({\mathscr{H}})$

$\displaystyle C_{U^{\xi}}(\varphi\otimes\overline{\psi})$ $\displaystyle=$
$\displaystyle\big{[}U^{\xi}\varphi,U^{\xi}\psi\big{]}_{{\mathscr{H}}}=\left[\int\limits_{\mathbb{R}^{d}}\varphi(s)d\xi(s),\int\limits_{\mathbb{R}^{d}}\psi(t)d\xi(t)\right]_{{\mathscr{H}}}=$
$\displaystyle=$
$\displaystyle\int\limits_{\mathbb{R}^{d}}\int\limits_{\mathbb{R}^{d}}\varphi(s)\overline{\psi(t)}d\big{(}\xi\otimes\xi\big{)}(s,t)=$
$\displaystyle=$
$\displaystyle\int\limits_{\mathbb{R}^{d}}\int\limits_{\mathbb{R}^{d}}\varphi(s)\overline{\psi(t)}d\tau_{\xi}(s,t),$

which by (5.7) gives the first relation in (5.12).

In particular for $\xi=\xi^{F}$ with $\xi^{F}$ given by (5.2), we have first
$U^{\xi}=U^{F}$, given by (5.5) and secondly, for each
$\varphi,\,\psi\in{\mathcal{D}}_{d}^{m}$

$\displaystyle C^{\tau_{\xi^{F}}}(\varphi\otimes\overline{\psi})$
$\displaystyle=$
$\displaystyle\int\limits_{\mathbb{R}^{d}}\int\limits_{\mathbb{R}^{d}}\varphi(s)\overline{\psi(t)}\Gamma_{F}(s,t)dsdt$
$\displaystyle=$ $\displaystyle
C^{\Gamma_{F}}(\varphi\otimes\overline{\psi}),$

which finally means the second relation from (5.12). ∎

Finally let’s observe that for the particular classes of stochastic mappings
considered in this Section a Wold type decomposition holds, which is closely
connected to the Wold decomposition given at the end of the previous Section.

For $F\in{\mathcal{E}}_{d}^{0}({\mathscr{H}})$ the _observable structure_
${\mathscr{H}}_{F}^{t_{0}}$ (respectively ${\mathscr{H}}_{(F)}^{t_{0}}$) is
the closed ${\mathcal{B}}(H)$-module (space) in ${\mathscr{H}}$ generated by
the set $\\{F(t),\ t\leq t_{o}\\}$, while for $\xi\in
fosvr{\mathcal{M}}_{d}({\mathscr{H}})$, ${\mathscr{H}}_{\xi}^{t_{0}}$,
(${\mathscr{H}}_{(\xi)}^{t_{0}}$) is the closed ${\mathcal{B}}(H)$-module
(space) in ${\mathscr{H}}$ generated by the set $\\{\xi(A),\
A\in\widetilde{\mathscr{B}}or({\mathbb{R}}^{d})\ :\ t\in A\Rightarrow t\leq
t_{0}\\}$.

Using the observable structures just defined one can easily infer
corresponding Wold decompositions for $F$ and $\xi$ as in Theorem 4.6:

(5.13) $F=F^{det}+F^{p}$

respectively

(5.14) $\xi=\xi^{det}+\xi^{p}.$

Since, as it is not hard to see, the above definitions of the observable
structures are coherent to the classical ones, i.e.

${\mathscr{H}}_{F}^{t_{0}}={\mathscr{H}}^{t_{0}}_{U^{F}}({\mathscr{H}}_{(F)}^{t_{0}}={\mathscr{H}}_{(U^{F})}^{t_{0}})\
\text{ and }\
{\mathscr{H}}_{\xi}^{t_{0}}={\mathscr{H}}_{U^{\xi}}^{t_{0}}({\mathscr{H}}_{(\xi)}^{t_{0}}={\mathscr{H}}_{(U^{\xi})}^{t_{0}}),$

we obtain the following coherence result for the three types of Wold
decomposition (Theorem 4.6, (5.13) and (5.14)).

######  Corollary 5.5.

The m.s.o.r.d.f.’s $F,\ \xi^{F},\ U^{F}$ as well as $\xi,\ U^{\xi}$ are
simultaneously (operator) deterministic, (operator) nondeterministic or
(operator) purely nondeterministic. Also the corresponding summands in the
Wold decomposition are the same, i.e.

$(U^{F})^{det}=U^{F^{det}},\quad(U^{F})^{p}=U^{F^{p}},$

as well as

$(U^{\xi})^{det}=U^{\xi^{det}}\quad(U^{\xi})^{p}=U^{\xi^{p}}.$

##  6\. Concluding remarks

Let’s mention that the invariance to the gramian unitary equivalences is an
important property not only for a pair of m.s.o. random distribution fields
with the same operator covariance distribution (Corollary 4.5), but for a
larger class of m.s.o. random distribution fields, namely those having the
property of operator stationarity. Such a study of stationarity, stationarily
cross correlatedness, the corresponding spectral representations, as well as
the periodically correlatedness are treated in [6] and [7]. An investigation
of some other classes of m.s.o. random distribution fields, which are
invariant to similarities (e.g. uniformly bounded linearly stationary m.s.o.
random distribution fields or other frameworks as in [30]), or which are
invariant to the actions of bounded ${\mathcal{B}}(H)$-linear operators
(especially harmonizable m.s.o. random distribution fields) and where the
concept of propagator, see [16] or [24], as well as some “intertwining”
properties of the Fourier transform with the covariance distribution mapping
(4.2), or with the covariance distribution bimeasure mappings (5.10), (5.11)
play important roles, will be conducted in some forthcoming papers.

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