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Karhunen-Loève expansion of Random Measures

Ricardo CARRIZO VERGARA
Université de Paris II Panthéon-Assas, Paris, France

<EMAIL_ADDRESS><EMAIL_ADDRESS>
February-Mars 2022

Abstract

We present an orthogonal expansion for real regular second-order finite random
measures over $\mathbb{R}^{d}$. Such expansion, which may be seen as a
Karhunen-Loève decomposition, consists in a series expansion of deterministic
real finite measures weighted by uncorrelated real random variables with
summable variances. The convergence of the series is in a mean-
square-$\mathcal{M}_{B}(\mathbb{R}^{d})^{*}$-weak∗ sense, with
$\mathcal{M}_{B}(\mathbb{R}^{d})$ being the space of bounded measurable
functions over $\mathbb{R}^{d}$. This is proven profiting the extra
requirement for a regular random measure that its covariance structure is
identified with a covariance measure over
$\mathbb{R}^{d}\times\mathbb{R}^{d}$. We also obtain a series decomposition of
the covariance measure which converges in a separately
$\mathcal{M}_{B}(\mathbb{R}^{d})^{*}$-weak${}^{*}-$total-variation sense. We
then obtain an analogous result for function-regulated regular random
measures.

Keywords Random Measure, Karhunen-Loève Expansion, Covariance Measure

##  1 Introduction

Karhunen-Loève expansions are an important tool for the analysis of stochastic
processes, both in theory and practice. The essential of such such expansions
consists on decompositions of the form

$X=\sum_{n\in\mathbb{N}}X_{n}f_{n},$ (1)

where $X$ is a (real) random variable with values in a (real) separable
Hilbert space $E$, $(f_{n})_{n\in\mathbb{N}}$ is an orthonormal basis of $E$,
and $(X_{n})_{n\in\mathbb{N}}$ is a sequence of uncorrelated random variables
with summable variances. Such expansion exists if
$\mathbb{E}(\|X\|^{2}_{E})<\infty$. The convergence of (1) is in such case
surely and in the sense of the mean-square error $\mathbb{E}(\|X-\sum_{j\leq
n}X_{j}f_{j}\|^{2}_{E})\xrightarrow[n\to\infty]{}0$. Similar expansions with
weaker convergence forms are obtained when $X$ is interpreted as a linear
application from $E$ to $L^{2}(\Omega,\mathcal{A},\mathbb{P})$ whose
covariance induces a continuous bi-linear form with finite trace. In such
case, the convergence of (1) is in the mean-square$-E^{*}$-weak∗ sense, that
is $\mathbb{E}(|X(e)-\sum_{j\leq
n}X_{j}(f_{j},e)_{E}|^{2})\xrightarrow[n\to\infty]{}0$ for every $e\in E$,
being $(\cdot,\cdot)_{E}$ the inner-product in $E$ (see Section 2). The
typical example used in most applications is the case of a mean-square
continuous stochastic process over a compact interval $(X(t))_{t\in[a,b]}$
which, through the study of its integral against square-integrable test-
functions, is interpreted as a process acting linearly over $L^{2}([a,b])$. We
refer to (Loève, 1978; Ghanem Spanos, 2003; Wang, 2008; Red-Horse Ghanem,
2009) for more information on the Karhunen-Loève expansion and its
applications.

Extensions of the principle (1) to non-Hilbert spaces has been worked out in
many manners. In every case, the vectors $(f_{n})_{n\in\mathbb{N}}$ can be
seen as members of some topological vector space $V$, not necessarily Hilbert.
Although one can always rely upon the Reproducing Kernel Hilbert Space induced
by the covariance structure of the process (see Aronszajn (1950) or Ghanem
Spanos (2003, Chapter 2)), which may play the role of the space $E$ in the
decomposition (1), it is often interesting to study other possible forms of
convergence of the series regarding the topology of $V$ instead of the one of
the Hilbert space $E$. The simplest example may be the one of the
aforementioned case of a mean-square continuous process over a compact
interval $(X(t))_{t\in[a,b]}$, where the series (1) consists (essentially) of
continuous functions $(f_{n})_{n\in\mathbb{N}}$, and for which, due to
Mercer’s Theorem, the convergence is in a uniform$-$mean-square sense (Loève,
1978, Section 37.5). In the setting of Gaussian processes, Bay Croix (2017)
studied Karhunen-Loève expansions of processes having values in separable
Banach spaces, Rajput (1972) studied the case of separable Fréchet spaces, and
Peccati Pycke (2010) studied expansions for processes defined over compact
topological groups. In Meidan (1979, Theorem 9) an orthogonal expansion for
Generalized Stochastic Processes, that is, random distributions, is presented
over compact subsets of $\mathbb{R}^{d}$. A case which is particularly
interesting to our aims is the one studied in (Carrizo Vergara, 2021), where
the studied process $X$ is a tempered random distribution over
$\mathbb{R}^{d}$, conceived as a continuous and linear application from the
Schwartz space $\mathscr{S}(\mathbb{R}^{d})$ to
$L^{2}(\Omega,\mathcal{A},\mathbb{P})$. For such $X$, it is there proven that
an orthogonal representation (1) holds with $(f_{n})_{n\in\mathbb{N}}$ being a
sequence of (essentially) linearly independent tempered distributions, and
with the convergence interpreted in a mean-
square$-\mathscr{S}^{\prime}(\mathbb{R}^{d})$-weak∗ sense.

In this work we propose a Karhunen-Loève expansion of a particular kind of
process $M$ which is a real second-order finite random measure over
$\mathbb{R}^{d}$ with a measure covariance structure (here called regular
random measure). We prove that such process can be decomposed in the form

$M=\sum_{n\in\mathbb{N}}X_{n}\mu_{n},$ (2)

where $(X_{n})_{n\in\mathbb{N}}$ is a sequence of uncorrelated random
variables whose variances form a convergent series, and
$(\mu_{n})_{n\in\mathbb{N}}$ is a sequence of real finite measures over
$\mathbb{R}^{d}$ which are essentially linearly independent. The convergence
of (2) is in the sense

$\mathbb{E}\left(\left|\langle M,\varphi\rangle-\sum_{j\leq
n}X_{j}\langle\mu_{j},\varphi\rangle\right|^{2}\right)\xrightarrow[n\to\infty]{}0,$
(3)

for every $\varphi$ measurable and bounded. The construction of the measures
$(\mu_{n})_{n\in\mathbb{N}}$ is done in a very similar manner to the
construction of the expansion for tempered random distributions done in
(Carrizo Vergara, 2021). We apply a regularising anti-derivative operator to
the random measure $M$ in order to obtain a mean-square continuous stochastic
process. This process has a traceable Karhunen-Loève expansion with respect to
$L^{2}(\mathbb{R}^{d},\nu)$, being $\nu$ a conveniently selected
(independently of $M$) measure. We then differentiate the terms of the
expansion in order to obtain the distributions $(\mu_{n})_{n\in\mathbb{N}}$
which turn out to be measures (for $n$ such that
$\sigma_{X_{n}}^{2}:=\mathbb{V}ar(X_{n})>0$, which are the important cases).
This also guarantees a convergence of the form (3) for $\varphi$ smooth and
compactly supported. The stronger case with $\varphi$ measurable and bounded
is obtained through a convenient application of Lusin’s Theorem. For the
covariance measure $C_{M}$ of $M$, which is a positive-semidefinite measure
over $\mathbb{R}^{d}\times\mathbb{R}^{d}$, it holds

$C_{M}=\sum_{n\in\mathbb{N}}\sigma_{X_{n}}^{2}\mu_{n}\otimes\mu_{n}.$ (4)

The mode of convergence of this series is in total-variation in one component
and with respect to a measurable and bounded test-function in the other
component. This result is then extended to the case of function-regulated
random measures, that is, the case where there exists a positive locally
bounded function $f$ such that $\frac{1}{f}M$ is a finite random measure.

To the knowledge of the author, an expansion such as (2) is not known in the
generality here proposed. It is known, for instance, that a White Noise can be
decomposed using any orthonormal basis of $L^{2}(\mathbb{R}^{d})$ as measures
$(\mu_{n})_{n\in\mathbb{N}}$ in (2) obtaining a non-traceable expansion. Such
expansion can be anyway re-transformed into a traceable one if the measures
and random variables are multiplied by convenient positive reciprocal numbers.
Similar arguments can be done in the case of orthogonal random measures by
using an orthonormal basis of $L^{2}(\mathbb{R}^{d},\mu)$, $\mu$ being the
controlling measure (see the end of Section 4 for our definition of orthogonal
random measures; we refer to (Passeggeri, 2020; Kingman, 1967) and (Schilling,
2016, Chapter 10) as authors using orthogonal random measures in stronger
senses: independently scattered or with almost-surely $\sigma-$additivity
properties). However, an expansion for a general function-controlled regular
random measure is not broadly known. The key extra supposition which allows to
obtain the result is that the covariance structure of $M$ is identified to a
covariance measure over $\mathbb{R}^{d}\times\mathbb{R}^{d}$. In general, the
covariance structure of a $L^{2}(\Omega,\mathcal{A},\mathbb{P})$-valued
measure over $\mathbb{R}^{d}$ has the structure of a bi-measure, which is not
always possible to identify to a measure (Rao, 2012). This extra regularity
property allows to use different results on $M$ and its covariance $C_{M}$,
such as the use of the total-variation measure $|C_{M}|$ and a Stochastic
Fubini Theorem. We remark that this does not imply that $M$ is a strictly
speaking random measure, in the sense that $M$ does not necessarily take
values in a space of measures, nor almost-surely.

In Section 2 we introduce the Karhunen-Loève expansion over Hilbert spaces and
we make precise which form of expansion and convergence we will use. In
Section 3.2 we introduce random measures. After setting our notation of
measure-theoretical objects, we introduce our concept of regular random
measures. We present some properties and result regarding those, including a
simplified version of a Stochastic Fubini Theorem 3.2. We also present the
properties of an anti-derivative regularising operator acting over finite
deterministic and random measures. In Section 4 we present the main Theorem
4.1 of expansion of finite regular random measures together with its proof.
The expansion of function-regulated random measures is presented in Theorem
4.2 as a corollary of Theorem 4.1. We end in Section 5 with some concluding
remarks.

We set up some notation. We denote $\mathbf{1}_{A}$ the indicator function of
the set $A$. The symbol $\|\cdot\|_{\infty}$ denotes the supremum norm, and it
is used for real functions. $\mathscr{D}(\mathbb{R}^{d})$ denotes the space of
(real) smooth compactly supported test-functions over $\mathbb{R}^{d}$
typically used in Distribution Theory. The abbreviation LDCT and CSI stand for
Lebesgue Dominated Convergence Theorem and Cauchy-Schwarz Inequality,
respectively. All the random variables considered in this work are supposed to
be defined over the same probability space $(\Omega,\mathcal{A},\mathbb{P})$
and to have zero mean, without loss of generality. A stochastic process is
understood as a family of random variables indexed by an arbitrary non-empty
set. We basically work with second-order properties, proper of mean-square
analysis. Hence, we do not make direct suppositions of the regularity of the
trajectories of the processes used. The most part of results we use,
particularly the definition of stochastic integrals, are always considering
limits in a mean-square sense. We do not precise the laws of the random
variables involved (the Gaussian case is a particular one).

##  2 Karhunen-Loève Expansion

We shall use a weak form of Karhunen-Loève expansions. The principle is the
following. Let $E$ be a real separable Hilbert space, with inner-product
$(\cdot,\cdot)_{E}$ and norm $\|\cdot\|_{E}$. Let $X:E\to
L^{2}(\Omega,\mathcal{A},\mathbb{P})$ be a linear and continuous real mapping
satisfying that there exists an orthonormal basis
$(e_{n})_{n\in\mathbb{N}}\subset E$ such that

$\sum_{n\in\mathbb{N}}\mathbb{E}(|X(e_{n})|^{2})<\infty.$ (5)

If this holds, we say that $X$ has a traceable Karhunen-Loève expansion with
respect to $E$. This because in such case there exist an orthonormal basis of
$E$, say $(f_{n})_{n\in\mathbb{N}}$, and a sequence of uncorrelated random
variables $(X_{n})_{n\in\mathbb{N}}$ with summable variances, such that

$X(e)=\sum_{n\in\mathbb{N}}X_{n}\left(f_{n},e\right)_{E},\quad\forall e\in E,$
(6)

the convergence of the series being considered in a mean-square sense. The
basis $(f_{n})_{n\in\mathbb{N}}$ consists of the eigenvectors of the
covariance operator induced by the covariance structure of $X$: if we denote
$K_{X}:E\times E\to\mathbb{R}$ the covariance Kernel of $X$, that is

$K_{X}(e,f)=\mathbb{E}(X(e)X(f)),$ (7)

then $K_{X}$ is bilinear, positive-semidefinite and continuous (since $X$ is
continuous). By Riesz Representation, for every $e\in E$ there exists an
element $Q_{X}(e)\in E$ such that

$K_{X}(e,f)=\left(Q_{X}(e),f\right)_{E},\quad\forall f\in E.$ (8)

The so-induced operator $Q_{X}:E\to E$ is called the covariance operator of
$X$. This operator is linear, continuous, positive-semidefinite, and by (5) it
is also trace-class (Reed Simon, 1980, Theorem VI.18). Hence, it has a
spectral decomposition in an orthonormal basis of eigenvectors
$(f_{n})_{n\in\mathbb{N}}$, with corresponding positive eigenvalues
$(\sigma_{X_{n}}^{2})_{n\in\mathbb{N}}$ which form a convergent series (Reed
Simon, 1980, Theorem VI.21). The random variables $(X_{n})_{n\in\mathbb{N}}$
are given by $X_{n}:=X(f_{n})$, for which we have
$\mathbb{C}ov(X_{n},X_{m})=\sigma_{X_{n}}^{2}\delta_{n,m}$.

We will mainly apply this principle to the following case. Consider
$(U(x))_{x\in\mathbb{R}^{d}}$ a real and mean-square continuous stochastic
process over $\mathbb{R}^{d}$. Consider $C_{U}(x,y)=\mathbb{E}(U(x)U(y))$ its
covariance function, which is continuous over
$\mathbb{R}^{d}\times\mathbb{R}^{d}$. Let $\nu$ be a positive finite measure
over $\mathbb{R}^{d}$ such that

$\int_{\mathbb{R}^{d}}C_{U}(x,x)d\nu(x)<\infty.$ (9)

Using CSI and the positive-semidefiniteness of $C_{U}$, one proves

$\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|C_{U}(x,y)||\varphi(x)||\phi(y)|d(\nu\otimes\nu)(x,y)<\infty,\quad\forall\varphi,\phi\in
L^{2}(\mathbb{R}^{d},\nu).$ (10)

It is known (Soong TT, 1973, Theorem 4.5.2) that this condition allows to
properly define stochastic integrals of the form

$\int_{\mathbb{R}^{d}}U(x)\varphi(x)d\nu(x),\quad\forall\varphi\in
L^{2}(\mathbb{R}^{d},\nu).$ (11)

Hence, one can re-define $U$ as a process indexed by functions in the
separable Hilbert space $L^{2}(\mathbb{R}^{d},\nu)$, setting

$\tilde{U}(\varphi):=\int_{\mathbb{R}^{d}}U(x)\varphi(x)d\nu(x),\quad\forall\varphi\in
L^{2}(\mathbb{R}^{d},\nu).$ (12)

The so-defined application $\tilde{U}:L^{2}(\mathbb{R}^{d},\nu)\to
L^{2}(\Omega,\mathcal{A},\mathbb{P})$ is continuous. The covariance operator
$Q_{\tilde{U}}:L^{2}(\mathbb{R}^{d},\nu)\to L^{2}(\mathbb{R}^{d},\nu)$ is
given by

$Q_{\tilde{U}}(\varphi)=\int_{\mathbb{R}^{d}}C_{U}(\cdot,y)\varphi(y)d\nu(y),$
(13)

which by (9) is trace-class (Brislawn, 1991). $\tilde{U}$ has then a traceable
Karhunen-Loève expansion with respect to $L^{2}(\mathbb{R}^{d},\nu)$:

$\int_{\mathbb{R}^{d}}U(x)\varphi(x)d\nu(x)=\sum_{n\in\mathbb{N}}X_{n}\langle\nu,\varphi
f_{n}\rangle,\quad\forall\varphi\in L^{2}(\mathbb{R}^{d},\nu),$ (14)

with $(f_{n})_{n\in\mathbb{N}}$ the orthonormal basis of eigenfunctions of
$Q_{\tilde{U}}$ and

$X_{n}=\int_{\mathbb{R}^{d}}U(x)f_{n}(x)d\nu(x),\quad\forall n\in\mathbb{N}.$
(15)

##  3 Random measures

###  3.1 Reminders on measures over $\mathbb{R}^{d}$

Let us recall and make precise our notation of measure-theoretical concepts
for the Euclidean space. We will always work in the context of the measurable
space $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$, being
$\mathcal{B}(\mathbb{R}^{d})$ the Borel $\sigma$-algebra. We denote
$\mathcal{B}_{B}(\mathbb{R}^{d})$ the $\delta$-ring of bounded Borel subsets
of $\mathbb{R}^{d}$.

By a measure over $\mathbb{R}^{d}$, we mean a real application
$\mu:\mathcal{B}_{B}(\mathbb{R}^{d})\to\mathbb{R}$ which is $\sigma$-additive
over $\mathcal{B}_{B}(\mathbb{R}^{d})$. This implies that $\mu$ is locally-
finite, but $\mu$ may not be defined over the whole Borel $\sigma$-algebra
(some authors use the term pre-measure for this object). $\mu$ is called
positive if it takes only non-negative values. For a real measure over $\mu$
over $\mathbb{R}^{d}$, its total-variation measure is defined as

$|\mu|(A):=\sup\\{\sum_{n\in\mathbb{N}}|\mu(E_{n})|\
\big{|}\big{(}E_{n})_{n\in\mathbb{N}}\subset\mathcal{B}_{B}(\mathbb{R}^{d})\hbox{
partition of }A\\},\quad\forall A\in\mathcal{B}_{B}(\mathbb{R}^{d}),$ (16)

which is the smallest positive measure such that $|\mu(A)|\leq|\mu|(A)$ for
all $A\in\mathcal{B}_{B}(\mathbb{R}^{d})$ (Rudin, 1987, Chapter 6). This
measure always exists and it is locally-finite. If $|\mu|$ can be extended
finitely and $\sigma$-additively to the whole Borel $\sigma$-algebra (hence
$|\mu|(\mathbb{R}^{d})<\infty$), then $\mu$ is said to be finite.

The space of (real) measurable functions over $\mathbb{R}^{d}$ is denoted
$\mathcal{M}(\mathbb{R}^{d})$. We denote $\mathcal{M}_{B}(\mathbb{R}^{d})$ the
space of bounded measurable functions and $\mathcal{M}_{B,c}(\mathbb{R}^{d})$
the space of bounded measurable functions with compact support. A function
$\mathcal{M}(\mathbb{R}^{d})$ is said to be integrable with respect to $\mu$
if $|f|$ is Lebesgue integrable with respect to $|\mu|$ over $\mathbb{R}^{d}$.
In such case, we use the notation

$\langle\mu,f\rangle:=\int_{\mathbb{R}^{d}}f\mu=\int_{\mathbb{R}^{d}}f(x)d\mu(x).$
(17)

We remark that the total variation measure $|\mu|$ can also be expressed as

$|\mu|(A)=\sup_{\varphi\in\mathcal{M}(\mathbb{R}^{d}),|\varphi|=\mathbf{1}_{A}}|\langle\mu,\varphi\rangle|,\quad\forall
A\in\mathcal{B}_{B}(\mathbb{R}^{d}).$ (18)

We recall the useful Lusin’s Theorem, applied over the space $\mathbb{R}^{d}$
(Folland, 1999, Theorem 7.10):

######  Theorem 3.1 (Lusin).

Let $\mu$ be a finite measure over $\mathbb{R}^{d}$. Let
$\psi\in\mathcal{M}(\mathbb{R}^{d})$. Then, for every $\epsilon>0$ there
exists a closed set $E\subset\mathbb{R}^{d}$ such that $\psi$ is continuous
over $E$ and $|\mu|(E^{c})<\epsilon$.

###  3.2 Random measures and properties

######  Definition 3.1.

A regular second-order random measure (from now on regular random measure)
over $\mathbb{R}^{d}$ is a stochastic process indexed by the bounded Borel
sets, $(M(A))_{A\in\mathcal{B}_{B}(\mathbb{R}^{d})}\subset
L^{2}(\Omega,\mathcal{A},\mathbb{P})$, such that there exists a real measure
$C_{M}$ over $\mathbb{R}^{d}\times\mathbb{R}^{d}$ such that

$\mathbb{E}(M(A)M(B))=C_{M}(A\times B),\quad\forall
A,B\in\mathcal{B}_{B}(\mathbb{R}^{d}).$ (19)

It is not difficult to prove that for such a process $M$ we have a
$\sigma$-additivity property

$M\left(\bigcup_{n\in\mathbb{N}}A_{n}\right)=\sum_{n\in\mathbb{N}}M(A_{n}),$
(20)

for every sequence of mutually disjoint sets
$(A_{n})_{n\in\mathbb{N}}\subset\mathcal{B}_{B}(\mathbb{R}^{d})$ such that
$\bigcup_{n\in\mathbb{N}}A_{n}$ is bounded, the series being taken in a mean-
square sense. Hence, $M$ is a $L^{2}(\Omega,\mathcal{A},\mathbb{P})$-valued
locally finite measure over $\mathbb{R}^{d}$. The extra adjective regular is
added because of the identification of the covariance structure of $M$ to the
covariance measure $C_{M}$. In general, a
$L^{2}(\Omega,\mathcal{A},\mathbb{P})$-valued locally finite measure over
$\mathbb{R}^{d}$ does not necessarily have a measure over
$\mathbb{R}^{d}\times\mathbb{R}^{d}$ describing its covariance structure: the
application $(A,B)\mapsto\mathbb{E}(M(A)M(B))$ defines a bi-measure which is
not necessarily identified to a measure. See (Rao, 2012, Chapter 2, Example 2)
for a counterexample.

It is clear that covariance measures are symmetric in the sense $C_{M}(A\times
B)=C_{M}(B\times A)$. It is also possible to verify that $|C_{M}|$ is also a
symmetric measure. Covariance measures are also positive-semidefinite in the
sense

$\langle C_{M},\varphi\otimes\varphi\rangle\geq
0,\quad\forall\varphi\in\mathcal{M}_{B,c}(\mathbb{R}^{d}).$ (21)

Conversely, every symmetric measure over $\mathbb{R}^{d}\times\mathbb{R}^{d}$
satisfying (21) is the covariance measure of a regular random measure
(Kolmogorov Extension Theorem).

######  Definition 3.2.

A regular random measure $M$ over $\mathbb{R}^{d}$ is said to be finite if its
covariance measure $C_{M}$ is finite.

When $M$ is finite, its definition can be extended uniquely, finitely and
$\sigma$-additively to the whole Borel $\sigma$-algebra
$\mathcal{B}(\mathbb{R}^{d})$, the random variable $M(\mathbb{R}^{d})$ having
finite variance.

If $\varphi\in\mathcal{M}(\mathbb{R}^{d})$ is such that

$\langle|C_{M}|,|\varphi|\otimes|\varphi|\rangle<\infty,$ (22)

then the stochastic integral

$\langle M,\varphi\rangle:=\int_{\mathbb{R}^{d}}\varphi(x)dM(x)$ (23)

can be defined as a random variable in $L^{2}(\Omega,\mathcal{A},\mathbb{P})$.
This is just an example of the Dunford-Schwartz (or Bochner) integral of
$\varphi$ with respect to the $L^{2}(\Omega,\mathcal{A},\mathbb{P})$-valued
measure $M$ (see (Rao, 2012, Chapter 2) for an effective introduction,
(Dunford Schwartz, 1958, Section IV.10) for the details, and (Carrizo Vergara,
2018, Proposition 3.3.1) for the sufficiency of condition (22)). If $\varphi$
and $\phi$ satisfy (22), then

$\mathbb{E}\left(\langle M,\varphi\rangle\langle M,\phi\rangle\right)=\langle
C_{M},\varphi\otimes\phi\rangle.$ (24)

Of course, every function in $\mathcal{M}_{B}(\mathbb{R}^{d})$ is integrable
in this sense with respect to any finite regular random measure, while every
function in $\mathcal{M}_{B,c}(\mathbb{R}^{d})$ is integrable with respect to
any regular random measure.

The next Theorem is a simplified form of what we could call a stochastic
Fubini Theorem. Its proof is given in Appendix A.

######  Theorem 3.2 (Stochastic Fubini).

Let $M$ be a regular random measure over $\mathbb{R}^{d}$ with covariance
measure $C_{M}$ and let $\mu$ be a (deterministic) finite measure over
$\mathbb{R}^{m}$. Let $\psi\in\mathcal{M}(\mathbb{R}^{d}\times\mathbb{R}^{m})$
be such that

  1. (i)

$\displaystyle\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{R}^{m}\times\mathbb{R}^{m}}|\psi(x,u)||\psi(y,v)|d(|C_{M}|\otimes|\mu|\otimes|\mu|)(x,y,u,v)<\infty.$

  2. (ii)

The function

$(u,v)\mapsto\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\psi(x,u)\psi(y,v)dC_{M}(x,y),$
(25)

which is defined outside a $|\mu|\otimes|\mu|$-null set, is continuous over a
set of the form $E\times E$ with $E\in\mathcal{B}(\mathbb{R}^{m})$ such that
$|\mu|(E^{c})=0$.

Then,

$\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{m}}\psi(x,u)d\mu(u)dM(x)=\int_{\mathbb{R}^{m}}\int_{\mathbb{R}^{d}}\psi(x,u)dM(x)d\mu(u).$
(26)

As the reader my notice, we use stochastic processes which are random measures
in a purely mean-square sense. That means that $M$ is not necessarily a
process with measure sample paths, that is, a random variable taking values in
a space of measures. Nor is it necessarily true that the process $M$ has
almost-surely measure-sample paths, or that it has a modification with measure
sample-paths. For those kinds of random measures, the definition of a
stochastic integral and stochastic Fubini theorem are relatively simple
results to obtain, since we can rely upon the deterministic theory. See the
big treaty (Kallenberg, 2017) for more information. As an important counter-
example, a Gaussian White Noise over $\mathbb{R}^{d}$, denoted $W$, which is a
regular random measure such that $C_{W}(A\times B)=|A\cap B|$, $|\cdot|$
standing for the Lebesgue measure, cannot be conceived as a process having a
modification with measure-sample paths. See (Dalang ., 2009, Example 3.16) for
such case, (Horowitz, 1986) for the general case of Gaussian measure-valued
random variables, and (Kingman, 1967) for the case of orthogonal
(independently-scattered) random measures.

###  3.3 Regularising operator

For non-trivial results exposed in this Section, we give their justifications
in Appendix B.

For a deterministic finite measure $\mu$ over $\mathbb{R}^{d}$, we define the
function $\mathcal{O}(\mu):\mathbb{R}^{d}\to\mathbb{R}$ as

$\mathcal{O}(\mu)(\vec{x}):=\int^{\vec{x}}_{0}\mu((-\infty,\vec{u}])d\vec{u},\quad\forall\vec{x}\in\mathbb{R}^{d},$
(27)

where we have used the abbreviated notations

$\int^{\vec{x}}_{0}(\cdot)\
d\vec{u}:=\int_{0}^{x_{1}}\int_{0}^{x_{2}}...\int_{0}^{x_{d}}(\cdot)\
du_{d}...du_{2}du_{1}$ (28)

and

$(-\infty,\vec{x}]:=(-\infty,x_{1}]\times(-\infty,x_{2}]\times...\times(-\infty,x_{d}],$
(29)

for every $\vec{x}=(x_{1},...,x_{d})\in\mathbb{R}^{d}$. Since $\mu$ is finite,
the function $\vec{u}\mapsto\mu((-\infty,\vec{u}])$ is measurable and bounded
by $|\mu|(\mathbb{R}^{d})$. The integral (27) is then well-defined, and the
resulting function $\vec{x}\mapsto\mathcal{O}(\mu)(\vec{x})$ is actually
continuous over $\mathbb{R}^{d}$. The next bound is easy to obtain:

$|\mathcal{O}(\mu)(\vec{x})|\leq|x_{1}|...|x_{d}||\mu|(\mathbb{R}^{d}),\quad\forall\vec{x}=(x_{1},...,x_{d})\in\mathbb{R}^{d}.$
(30)

Finally and more crucially, we have

$\frac{\partial^{2d}\mathcal{O}(\mu)}{\partial x_{1}^{2}...\partial
x_{d}^{2}}=\mu$ (31)

in distributional sense over $\mathbb{R}^{d}$, that is

$\int_{\mathbb{R}^{d}}\mathcal{O}(\mu)(x)\frac{\partial^{2d}\varphi}{\partial
x_{1}^{2}...\partial
x_{d}^{2}}(x)dx=\int_{\mathbb{R}^{d}}\varphi(x)d\mu(x),\quad\forall\varphi\in\mathscr{D}(\mathbb{R}^{d}),$
(32)

which can be proven using Fubini Theorem or through convolution arguments.

Let us now apply $\mathcal{O}$ to a finite regular random measure $M$. The
application $\vec{u}\mapsto M((-\infty,\vec{u}])$ determines a stochastic
process over $\mathbb{R}^{d}$ which has the particularity of having a bounded
covariance and of being mean-square continuous outside a set of null Lebesgue
measure. Hence, the stochastic integral

$\mathcal{O}(M)(\vec{x}):=\int_{0}^{\vec{x}}M((-\infty,\vec{u}])d\vec{u}$ (33)

is well-defined through Riemann-alike approximations. This process has
covariance function

$C_{\mathcal{O}(M)}(\vec{x},\vec{y})=\int_{0}^{\vec{x}}\int_{0}^{\vec{y}}C_{M}((-\infty,\vec{u}]\times(-\infty,\vec{v}])d\vec{v}d\vec{u},$
(34)

which is a continuous function over $\mathbb{R}^{d}\times\mathbb{R}^{d}$ (it
is actually the function $\mathcal{O}(C_{M})$ interpreting $\mathcal{O}$ as an
operator acting over finite measures over
$\mathbb{R}^{d}\times\mathbb{R}^{d}$). Hence $\mathcal{O}(M)$ is mean-square
continuous. In addition one has the bound

$|C_{\mathcal{O}(M)}(\vec{x},\vec{y})|\leq|x_{1}|...|x_{d}||y_{1}|...|y_{d}||C_{M}|(\mathbb{R}^{d}\times\mathbb{R}^{d}),\quad\forall\vec{x},\vec{y}\in\mathbb{R}^{d}.$
(35)

Finally, an application of Stochastic Fubini Theorem 3.2 allows to conclude

$\frac{\partial^{2d}\mathcal{O}(M)}{\partial x_{1}^{2}...\partial
x_{d}^{2}}=M$ (36)

in distributional sense over $\mathbb{R}^{d}$, that is, we have the equality
between the stochastic integrals

$\int_{\mathbb{R}^{d}}\mathcal{O}(M)(x)\frac{\partial^{2d}\varphi}{\partial
x_{1}^{2}...\partial
x_{d}^{2}}(x)dx=\int_{\mathbb{R}^{d}}\varphi(x)dM(x),\quad\forall\varphi\in\mathscr{D}(\mathbb{R}^{d}).$
(37)

##  4 Expansion of random measures

We give first a result of a Karhunen-Loève expansion for regular finite random
measures.

######  Theorem 4.1 (Karhunen-Loève expansion of finite random measures).

let $M$ be a finite regular random measure over $\mathbb{R}^{d}$. Then, there
exists a sequence of pairwise uncorrelated random variables with summable
variances $(X_{n})_{n\in\mathbb{N}}$, and a sequence of finite measures over
$\mathbb{R}^{d}$, $(\mu_{n})_{n\in\mathbb{N}}$ such that

$\langle
M,\varphi\rangle=\sum_{j\in\mathbb{N}}X_{j}\langle\mu_{j},\varphi\rangle,\quad\forall\varphi\in\mathcal{M}_{B}(\mathbb{R}^{d}),$
(38)

with the series being considered in a mean-square sense. In addition, if
$C_{M}$ is the covariance measure of $M$, then,

$C_{M}=\sum_{j\in\mathbb{N}}\sigma_{X_{j}}^{2}\mu_{j}\otimes\mu_{j},$ (39)

where $\sigma_{X_{n}}^{2}=\mathbb{V}ar(X_{n})$ for all $n\in\mathbb{N}$. The
convergence of the series is in a separately
$\mathcal{M}_{B}(\mathbb{R}^{d})^{*}$-weak${}^{*}-$total-variation sense, that
is,

$\left|\langle C_{M},\varphi\otimes(\cdot)\rangle-\sum_{j\leq
n}\sigma_{X_{j}}^{2}\langle\mu_{j},\varphi\rangle\mu_{j}\right|(\mathbb{R}^{d})\xrightarrow[n\to\infty]{}0\quad\forall\varphi\in\mathcal{M}_{B}(\mathbb{R}^{d}).$
(40)

The reader will recognize very similar arguments to the classical proof of the
Karhunen-Loève expansion for mean-square continuous stochastic process over
compact intervals, see for example (Giambartolomei, 2015) for a complete and
detailed non-specialist oriented exposition. In such case, Dini’s Theorem
allows to conclude a uniform-mean-square convergence. Such role is played in
our case by Lusin’s Theorem.

Proof: We begin with the construction of the random variables
$(X_{n})_{n\in\mathbb{N}}$ and the measures $(\mu_{n})_{n\in\mathbb{N}}$. For
that, we consider the function $p:\mathbb{R}^{d}\to\mathbb{R}^{+}$ given by

$p(\vec{x})=\prod_{j=1}^{d}(1+|x_{j}|^{2})^{2},\quad\vec{x}=(x_{1},...,x_{d})\in\mathbb{R}^{d},$
(41)

which is strictly positive and locally bounded. We also set the finite measure
over $\mathbb{R}^{d}$

$d\nu(x)=\frac{dx}{p(x)}.$ (42)

Now, we consider the stochastic process over $\mathbb{R}^{d}$

$U=\mathcal{O}(M).$ (43)

We have seen that this process is mean-square continuous and that it has the
properties presented in Section 3.3. From relation (35) we conclude

$\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}C_{U}(x,x)d\nu(x)\leq|C_{M}|(\mathbb{R}^{d}\times\mathbb{R}^{d})\left(\int_{\mathbb{R}}\frac{t^{2}}{(1+t^{2})^{2}}dt\right)^{d}<\infty.$
(44)

Hence, as stated in Section 2, the process $U$ has a traceable Karhunen-Loève
expansion with respect to the Hilbert space $L^{2}(\mathbb{R}^{d},\nu)$,
having thus

$\int_{\mathbb{R}^{d}}U(x)\varphi(x)d\nu(x)=\sum_{j\in\mathbb{N}}X_{j}\langle\nu,f_{j}\varphi\rangle,\quad\forall\varphi\in
L^{2}(\mathbb{R}^{d},\nu),$ (45)

being $(f_{j})_{j\in\mathbb{N}}$ the orthonormal basis of
$L^{2}(\mathbb{R}^{d},\nu)$ given by the eigenfunctions of the trace-class
covariance operator (13), and the random variables $(X_{j})_{j\in\mathbb{N}}$
are given by (15) and they are uncorrelated with variances
$(\sigma_{X_{n}}^{2})_{n\in\mathbb{N}}$ such that
$\sum_{n\in\mathbb{N}}\sigma_{X_{n}}^{2}<\infty$.

We remark that a function $f\in L^{2}(\mathbb{R}^{d},\nu)$ determines a
distribution over $\mathbb{R}^{d}$. This because $f$ is equivalent to the
locally finite measure $fp\nu$ in the sense that $\langle
f,\varphi\rangle=\langle fp\nu,\varphi\rangle$ for every
$\varphi\in\mathscr{D}(\mathbb{R}^{d})$ (the measure $p\nu$ is the Lebesgue
measure). To see that $fp\nu$ is locally finite, consider any
$A\in\mathcal{B}_{B}(\mathbb{R}^{d})$ and by CSI one has

$\int_{A}|f|pd\nu\leq\sqrt{\int_{A}p^{2}d\nu}\sqrt{\int_{A}|f|^{2}d\nu}\leq\sqrt{\int_{A}pdx}\|f\|_{L^{2}(\mathbb{R}^{d},\nu)}<\infty,$
(46)

the last expression being finite due to the local-boundedness of $p$. For
$\varphi\in\mathscr{D}(\mathbb{R}^{d})$, one has $\varphi p\in
L^{2}(\mathbb{R}^{d},\nu)$ hence from (45) we have

$\langle
U,\varphi\rangle=\int_{\mathbb{R}^{d}}U(x)\varphi(x)p(x)d\nu(x)=\sum_{j\in\mathbb{N}}X_{j}\langle\nu,f_{j}\varphi
p\rangle=\sum_{j\in\mathbb{N}}X_{j}\langle f_{j},\varphi\rangle.$ (47)

Considering the derivative relation (36), we conclude

$\displaystyle\langle M,\varphi\rangle$
$\displaystyle=\langle\frac{\partial^{2d}U}{\partial x_{1}^{2}...\partial
x_{d}^{2}},\varphi\rangle$ (48) $\displaystyle=\langle
U,\frac{\partial^{2d}\varphi}{\partial x_{1}^{2}...\partial x_{d}^{2}}\rangle$
$\displaystyle=\sum_{j\in\mathbb{N}}X_{j}\langle
f_{j},\frac{\partial^{2d}\varphi}{\partial x_{1}^{2}...\partial
x_{d}^{2}}\rangle$
$\displaystyle=\sum_{j\in\mathbb{N}}X_{j}\langle\frac{\partial^{2d}f_{j}}{\partial
x_{1}^{2}...\partial
x_{d}^{2}},\varphi\rangle,\quad\forall\varphi\in\mathscr{D}(\mathbb{R}^{d}).$

This indicates that we must study the distributions
$\frac{\partial^{2d}f_{j}}{\partial x_{1}^{2}...\partial x_{d}^{2}}$. We set
$\sigma_{X_{j}}^{2}:=\mathbb{V}ar(X_{j})$ and we consider the cases where
$\sigma_{X_{j}}^{2}>0$. From the eigenfunction condition and the covariance of
$C_{U}$ given by (34), we obtain

$\displaystyle f_{j}(\vec{x})$
$\displaystyle=\frac{1}{\sigma_{X_{j}}^{2}}\int_{\mathbb{R}^{d}}C_{U}(\vec{x},\vec{y})f_{j}(\vec{y})d\nu(\vec{y})$
(49)
$\displaystyle=\frac{1}{\sigma_{X_{j}}^{2}}\int_{\mathbb{R}^{d}}\int_{0}^{\vec{x}}\int_{0}^{\vec{y}}C_{M}((-\infty,\vec{u}]\times(-\infty,\vec{v}])d\vec{v}d\vec{u}f_{j}(\vec{y})d\nu(\vec{y})$
$\displaystyle=\frac{1}{\sigma_{X_{j}}^{2}}\int_{0}^{\vec{x}}\int_{\mathbb{R}^{d}}\int_{0}^{\vec{y}}C_{M}((-\infty,\vec{u}]\times(-\infty,\vec{v}])d\vec{v}f_{j}(\vec{y})d\nu(\vec{y})d\vec{u},$

where we have used the (deterministic) Fubini Theorem. Following this result,
we define the application $\mu_{j}:\mathcal{B}(\mathbb{R}^{d})\to\mathbb{R}$
as

$\mu_{j}(A):=\frac{1}{\sigma_{X_{j}}^{2}}\int_{\mathbb{R}^{d}}\int_{0}^{\vec{y}}C_{M}(A\times(-\infty,\vec{v}])d\vec{v}f_{j}(\vec{y})d\vec{y}.$
(50)

We remark that we can re-write $\mu_{j}$ as

$\mu_{j}(A)=\frac{1}{\sigma_{X_{j}}^{2}}\int_{\mathbb{R}^{d}}\mathcal{O}\big{(}C_{M}(A\times\cdot)\big{)}(\vec{y})f_{j}(\vec{y)}d\nu(\vec{y}),$
(51)

where $C_{M}(A\times\cdot)$ stands for the measure over $\mathbb{R}^{d}$ given
by $B\mapsto C_{M}(A\times B)$. Given the property (30) of the operator
$\mathcal{O}$, the function $\mathcal{O}\big{(}C_{M}(A\times\cdot)\big{)}$ is
in $L^{2}(\mathbb{R}^{d},\nu)$, hence the integral (51) is well-defined. It is
clear that $\mu_{j}$ is an additive function. In addition, one has

$|\mu_{j}(A)|\leq\frac{1}{\sigma_{X_{j}}^{2}}\underbrace{\int_{\mathbb{R}^{d}}|y_{1}|...|y_{d}||f_{j}(\vec{y})|d\nu(\vec{y})}_{<\infty\hbox{
since }\vec{y}\mapsto|y_{1}|...|y_{d}|\ \in\
L^{2}(\mathbb{R}^{d},\nu)}|C_{M}|(A\times\mathbb{R}^{d}).$ (52)

Since $|C_{M}|$ is a finite measure, if we take any sequence of Borel sets
$(A_{n})_{n\in\mathbb{N}}$ such that $A_{n}\searrow\emptyset$, we must have
$|C_{M}|(A_{n}\times\mathbb{R}^{d})\searrow 0$ and hence $|\mu_{j}(A_{n})|\to
0$. This proves that $\mu_{j}$ is a measure over $\mathbb{R}^{d}$ ans it is
also finite since $C_{M}$ is finite. In addition, from (49) we have

$f_{j}(\vec{x})=\int_{0}^{\vec{x}}\mu_{j}((-\infty,\vec{u}])d\vec{u}=\mathcal{O}(\mu_{j})(\vec{x}),$
(53)

from where we conclude

$\frac{\partial^{2d}f_{j}}{\partial x_{1}^{2}...\partial x_{d}^{2}}=\mu_{j}.$
(54)

For the case where $\sigma_{X_{j}}^{2}=0$, the distribution
$\frac{\partial^{2d}f_{j}}{\partial x_{1}^{2}...\partial x_{d}^{2}}$ is not
necessarily a measure, but such case is not really important since for such
$j$ we have $X_{j}=0$, so $\frac{\partial^{2d}f_{j}}{\partial
x_{1}^{2}...\partial x_{d}^{2}}$ does not really intervene in the expansion
(38). For simplicity, we will assume in the following that
$\sigma_{X_{j}}^{2}>0$ for all $j\in\mathbb{N}$ (for the case where the sum
(38) is finite we have nothing more to prove).

We have hence proven the expansion

$\langle
M,\varphi\rangle=\sum_{j\in\mathbb{N}}X_{j}\langle\mu_{j},\varphi\rangle,\quad\forall\varphi\in\mathscr{D}(\mathbb{R}^{d}).$
(55)

Now, the objective is to extend the result to
$\varphi\in\mathcal{M}_{B}(\mathbb{R}^{d})$. For that, let us define and
analyse the sequence of finite random measures

$M_{n}=\sum_{j\leq n}X_{j}\mu_{j},\quad n\in\mathbb{N}.$ (56)

Let us study the covariance of $M-M_{n}$. For that, we consider
$A,B\in\mathcal{B}(\mathbb{R}^{d})$. Then,

$\displaystyle\mathbb{E}\left(\big{(}M(A)-M_{n}(A)\big{)}\big{(}M(B)-M_{n}(B)\big{)}\right)=$
$\displaystyle C_{M}(A\times B)-\mathbb{E}(M(A)M_{n}(B))$ (57)
$\displaystyle-\mathbb{E}(M(B)M_{n}(A))+\mathbb{E}(M_{n}(A)M_{n}(B))$

The covariance $C_{M_{n}}$ is given by

$\displaystyle C_{M_{n}}(A\times B)$
$\displaystyle=\mathbb{E}\left(M_{n}(A)M_{n}(B)\right)$ (58)
$\displaystyle=\mathbb{E}\left(\sum_{j\leq n}\sum_{k\leq
n}X_{j}X_{k}\mu_{j}(A)\mu_{k}(B)\right)$ $\displaystyle=\sum_{j\leq
n}\sigma_{X_{j}}^{2}\mu_{j}(A)\mu_{j}(B),$

where we have used that
$\mathbb{E}(X_{j}X_{k})=\sigma_{X_{j}}^{2}\delta_{j,k}$. On the other hand,

$\displaystyle\mathbb{E}\left(M(A)M_{n}(B)\right)$
$\displaystyle=\mathbb{E}\left(\sum_{j\leq n}M(A)X_{j}\mu_{j}(B)\right)$ (59)
$\displaystyle=\sum_{j\leq n}\mathbb{E}\left(M(A)X_{j}\right)\mu_{j}(B).$

Let us compute $\mathbb{E}\left(M(A)X_{j}\right)$. For that we re-write the
expression (15) for $X_{j}$ as

$X_{j}=\int_{\mathbb{R}^{d}}\mathcal{O}(M)(\vec{y})f_{j}(\vec{y})d\vec{y}=\int_{\mathbb{R}^{d}}\int_{0}^{\vec{y}}\int_{\mathbb{R}^{d}}\mathbf{1}_{(-\infty,\vec{u}]}(\vec{s})dM(\vec{s})d\vec{u}f_{j}(\vec{y})d\nu(\vec{y}).$
(60)

We seek to write this integral as an integral with respect to $M$. For that,
we will use Stochastic Fubini Theorem 3.2. Consider for every
$\vec{y}\in\mathbb{R}^{d}$ the function
$\theta_{\vec{y}}:\mathbb{R}^{d}\to\\{-1,1\\}$ such that

$\int_{\mathbb{R}^{d}}\theta_{\vec{y}}(\vec{u})\varphi(\vec{u})d\vec{u}=\int_{0}^{\vec{y}}\varphi(\vec{u})d\vec{u},\quad\forall\varphi\in\mathcal{M}_{B,c}(\mathbb{R}^{d}).$
(61)

We can re-write the integral (60) as

$X_{j}=\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\mathbf{1}_{(-\infty,\vec{u}]}(\vec{s})dM(\vec{s})\theta_{\vec{y}}(\vec{u})f_{j}(\vec{y})d(\ell^{d}\otimes\nu)(\vec{u},\vec{y}),$
(62)

where we have denoted by $\ell^{d}$ the Lebesgue measure over
$\mathbb{R}^{d}$. We remark that the measure over
$\mathbb{R}^{d}\times\mathbb{R}^{d}$ given by
$d\mu(\vec{u},\vec{y}):=\theta_{\vec{y}}(\vec{u})f_{j}(\vec{y})d(\ell^{d}\otimes\nu)(\vec{u},\vec{y})$
is finite, since by Fubini

$\displaystyle\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|\theta_{\vec{y}}(\vec{u})f_{j}(\vec{y})|d(\ell^{d}\otimes\nu)(\vec{u},\vec{y})$
$\displaystyle=\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}|\theta_{\vec{y}}(\vec{u})|d\vec{u}|f_{j}(\vec{y})|d\nu(\vec{y})$
(63)
$\displaystyle\leq\int_{\mathbb{R}^{d}}|y_{1}|...|y_{d}||f_{j}(\vec{y})|d\nu(\vec{y})<\infty.$

If we consider the function
$\psi\in\mathcal{M}(\mathbb{R}^{d}\times(\mathbb{R}^{d}\times\mathbb{R}^{d}))$
given by

$\psi(\vec{s},(\vec{u},\vec{y}))=\mathbf{1}_{(-\infty,\vec{u}]}(\vec{s}),$
(64)

we see that condition (i) holds since the corresponding full integral equals

$\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|C_{M}|((-\infty,\vec{u}]\times(-\infty,\vec{v}])d|\mu|(\vec{u},\vec{y})d|\mu|(\vec{v},\vec{z}),$
(65)

which is finite since $|\mu|$ is finite and the function
$(\vec{u},\vec{v})\mapsto|C_{M}|((-\infty,\vec{u}]\times(-\infty,\vec{v}])$ is
bounded. In addition, condition (25) holds since $(\vec{u},\vec{v})\mapsto
C_{M}((-\infty,\vec{u}]\times(-\infty,\vec{v}])$ is continuous over a set of
the form $E\times E$ such that $\ell^{d}(E^{c})=0$, see those details in
Appendix B. Stochastic Fubini Theorem can then be applied, and we have thus

$\displaystyle X_{j}$
$\displaystyle=\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\mathbf{1}_{(-\infty,\vec{u}]}(\vec{s})\theta_{\vec{y}}(\vec{u})f_{j}(\vec{y})d(\ell^{d}\otimes\nu)(\vec{u},\vec{y})dM(\vec{s})$
(66)
$\displaystyle=\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\int_{0}^{\vec{y}}\mathbf{1}_{(-\infty,\vec{u}]}(\vec{s})d\vec{u}f_{j}(\vec{y})d\nu(\vec{y})dM(\vec{s}).$

Using formula (24) and the deterministic Fubini Theorem,

$\displaystyle\mathbb{E}\left(M(A)X_{j}\right)$
$\displaystyle=\mathbb{E}\left(\int_{\mathbb{R}^{d}}\mathbf{1}_{A}(\vec{t})dM(\vec{t})\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\int_{0}^{\vec{y}}\mathbf{1}_{(-\infty,\vec{u}]}(\vec{s})d\vec{u}f_{j}(\vec{y})d\nu(\vec{y})dM(\vec{s})\right)$
(67)
$\displaystyle=\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\mathbf{1}_{A}(\vec{t})\int_{\mathbb{R}^{d}}\int_{0}^{\vec{y}}\mathbf{1}_{(-\infty,\vec{u}]}(\vec{s})d\vec{u}f_{j}(\vec{y})d\nu(\vec{y})dC_{M}(\vec{t},\vec{s})$
$\displaystyle=\int_{\mathbb{R}^{d}}\int_{0}^{\vec{y}}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\mathbf{1}_{A}(\vec{t})\mathbf{1}_{(-\infty,\vec{u}]}(\vec{s})dC_{M}(\vec{t},\vec{s})d\vec{u}f_{j}(\vec{y})\nu(\vec{y})$
$\displaystyle=\int_{\mathbb{R}^{d}}\int_{0}^{\vec{y}}C_{M}(A\times(-\infty,\vec{u}])d\vec{u}f_{j}(\vec{y})d\nu(\vec{y})$
$\displaystyle=\int_{\mathbb{R}^{d}}\mathcal{O}\big{(}C_{M}(A\times\cdot)\big{)}(\vec{y})f_{j}(\vec{y})d\nu(\vec{y})$
$\displaystyle=\sigma_{X_{j}}^{2}\mu_{j}(A).$

We conclude

$\mathbb{E}\left(M(A)M_{n}(B)\right)=\sum_{j\leq
n}\sigma_{X_{j}}^{2}\mu_{j}(A)\mu_{j}(B)=C_{M_{n}}(A\times B).$ (68)

Of course formula (68) also holds when interchanging $A$ and $B$. Using this
in (57) we conclude that $C_{M-M_{n}}$ is the finite measure given by

$C_{M-M_{n}}(A\times B)=C_{M}(A\times B)-\sum_{j\leq
n}\sigma_{X_{j}}^{2}\mu_{j}(A)\mu_{j}(B),$ (69)

or equivalently,

$C_{M-M_{n}}=C_{M}-C_{M_{n}}=C_{M}-\sum_{j\leq
n}\sigma_{X_{j}}^{2}\mu_{j}\otimes\mu_{j}.$ (70)

Since $M-M_{n}$ is finite, this is equivalent to have for every
$\varphi,\phi\in\mathcal{M}_{B}(\mathbb{R}^{d})$,

$\langle C_{M-M_{n}},\varphi\otimes\phi\rangle=\langle
C_{M},\varphi\otimes\phi\rangle-\sum_{j\leq
n}\sigma_{X_{j}}^{2}\langle\mu_{j},\varphi\rangle\langle\mu_{j},\phi\rangle.$
(71)

$C_{M-M_{n}}$ is a covariance measure hence it is positive-semidefinite. Thus,
$\langle C_{M-M_{n}},\varphi\otimes\varphi\rangle\geq 0$ for every
$\varphi\in\mathcal{M}_{B}(\mathbb{R}^{d})$. This together with (71) allows to
conclude

$0\leq\langle C_{M_{n}},\varphi\otimes\varphi\rangle\leq\ \langle
C_{M},\varphi\otimes\varphi\rangle,\quad\forall
n\in\mathbb{N},\forall\varphi\in\mathcal{M}_{B}(\mathbb{R}^{d}).$ (72)

Then, by CSI,

$\displaystyle\sum_{j\leq
n}\sigma_{X_{j}}^{2}|\langle\mu_{j},\varphi\rangle||\langle\mu_{j},\phi\rangle|$
$\displaystyle\leq\sqrt{\sum_{j\leq
n}\sigma_{X_{j}}^{2}|\langle\mu_{j},\varphi\rangle|^{2}}\sqrt{\sum_{j\leq
n}\sigma_{X_{j}}^{2}|\langle\mu_{j},\phi\rangle|^{2}}$ (73)
$\displaystyle=\sqrt{\langle
C_{M_{n}},\varphi\otimes\varphi\rangle}\sqrt{\langle
C_{M_{n}},\phi\otimes\phi\rangle}$ $\displaystyle\leq\sqrt{\langle
C_{M},\varphi\otimes\varphi\rangle}\sqrt{\langle
C_{M},\phi\otimes\phi\rangle}<\infty.$

This proves that the series
$\sum_{j\in\mathbb{N}}\sigma_{X_{j}}^{2}\langle\mu_{j},\varphi\rangle\langle\mu_{j},\phi\rangle$
is absolutely convergent for every
$\varphi,\phi\in\mathcal{M}_{B}(\mathbb{R}^{d})$. We can hence define a
function
$\Lambda:\mathcal{M}_{B}(\mathbb{R}^{d})\times\mathcal{M}_{B}(\mathbb{R}^{d})\to\mathbb{R}$
as

$\Lambda(\varphi,\phi):=\sum_{j\in\mathbb{N}}\sigma_{X_{j}}^{2}\langle\mu_{j},\varphi\rangle\langle\mu_{j},\phi\rangle.$
(74)

$\Lambda$ is a bilinear positive-semidefinite function. It is clear that in
order to complete the proof of Theorem 4.1 we need only to prove

$\Lambda(\varphi,\phi)=\langle
C_{M},\varphi\otimes\phi\rangle,\quad\forall\varphi,\phi\in\mathcal{M}_{B}(\mathbb{R}^{d}).$
(75)

Using the positive-semidefiniteness of $C_{M-M_{n}}$ and CSI, one has that
(75) is equivalent to

$\Lambda(\varphi,\varphi)=\langle
C_{M},\varphi\otimes\varphi\rangle,\quad\forall\varphi\in\mathcal{M}_{B}(\mathbb{R}^{d}).$
(76)

The result (55) guarantees that (76) holds for
$\varphi\in\mathscr{D}(\mathbb{R}^{d})$. We will extend it to
$\varphi\in\mathcal{M}_{B}(\mathbb{R}^{d})$.

We begin by considering $\varphi$ of the form $\varphi=\mathbf{1}_{I}$, where
$I\subset\mathbb{R}^{d}$ is a rectangle, that is, $I=I_{1}\times...\times
I_{d}$, the set $I_{j}$ being an interval of $\mathbb{R}$ for every $j$. It is
well-known that in such case $\varphi$ can be approximated point-wise by a
sequence of functions in $\mathscr{D}(\mathbb{R}^{d})$, being them all
dominated by $\|\varphi\|_{\infty}$. We use hence LCDT considering the finite
measure $|C_{M}|$. Let $\epsilon>0$. Then, there exists a function
$\phi\in\mathscr{D}(\mathbb{R}^{d})$ having
$\|\phi\|_{\infty}\leq\|\varphi\|_{\infty}$, such that

$\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|\varphi-\phi|\otimes|\varphi-\phi|d|C_{M}|<\epsilon^{2}.$
(77)

We consider a typical application of triangular inequality in order to obtain

$\displaystyle\left|\Lambda(\varphi,\varphi)-\langle
C_{M},\varphi\otimes\varphi\rangle\right|\leq$
$\displaystyle\underbrace{\left|\Lambda(\varphi,\varphi)-\sum_{j\leq
n}\sigma_{X_{j}}^{2}|\langle\mu_{j},\varphi\rangle|^{2}\right|}_{(a)}$ (78)
$\displaystyle+\underbrace{\left|\sum_{j\leq
n}\sigma_{X_{j}}^{2}|\langle\mu_{j},\varphi\rangle|^{2}-\sum_{j\leq
n}\sigma_{X_{j}}^{2}|\langle\mu_{j},\phi\rangle|^{2}\right|}_{(b)}$
$\displaystyle+\underbrace{\left|\sum_{j\leq
n}\sigma_{X_{j}}^{2}|\langle\mu_{j},\phi\rangle|^{2}-\langle
C_{M},\phi\otimes\phi\rangle\right|}_{(c)}$
$\displaystyle+\underbrace{\left|\langle C_{M},\phi\otimes\phi\rangle-\langle
C_{M},\varphi\otimes\varphi\rangle\right|}_{(d)}.$

We will bound each one of the terms $(a),(b),(c),(d)$. Since
$\varphi\in\mathcal{M}_{B}(\mathbb{R}^{d})$, by definition of $\Lambda$ (74),
the term $(a)$ goes to $0$ as $n\to\infty$. The same idea goes for the term
$(c)$, since (76) holds for $\phi\in\mathscr{D}(\mathbb{R}^{d})$. There exists
hence $n_{0}$ such that if $n\geq n_{0}$, $\max\\{(a),(c)\\}<\epsilon$. On the
other hand, one has for $(b)$:

$\displaystyle(b)$ $\displaystyle=\left|\sum_{j\leq
n}\sigma_{X_{j}}^{2}\langle\mu_{j}\otimes\mu_{j},\varphi\otimes\varphi-\phi\otimes\phi\rangle\right|$
(79) $\displaystyle=\left|\sum_{j\leq
n}\sigma_{X_{j}}^{2}\langle\mu_{j}\otimes\mu_{j},\varphi\otimes(\varphi-\phi)+(\varphi-\phi)\otimes\phi\rangle\right|$
$\displaystyle\leq\left|\sum_{j\leq
n}\sigma_{X_{j}}^{2}\langle\mu_{j},\varphi\rangle\langle\mu_{j},\varphi-\phi\rangle\right|+\left|\sum_{j\leq
n}\sigma_{X_{j}}^{2}\langle\mu_{j},\varphi-\phi\rangle\langle\mu_{j},\phi\rangle\right|$

We use $\|\phi\|_{\infty}\leq\|\varphi\|_{\infty}$ and the bounds (73) to
obtain

$\displaystyle(b)$ $\displaystyle\leq\sqrt{\langle
C_{M},\varphi\otimes\varphi\rangle\langle
C_{M},(\varphi-\phi)\otimes(\varphi-\phi)\rangle}+\sqrt{\langle
C_{M},\phi\otimes\phi\rangle\langle
C_{M},(\varphi-\phi)\otimes(\varphi-\phi)\rangle}$ (80)
$\displaystyle\leq\left(\sqrt{|C_{M}|(\mathbb{R}^{d}\times\mathbb{R}^{d})\|\varphi\|_{\infty}^{2}}+\sqrt{|C_{M}|(\mathbb{R}^{d}\times\mathbb{R}^{d})\|\phi\|_{\infty}^{2}}\right)\underbrace{\sqrt{\langle|C_{M}|,|\varphi-\phi|\otimes|\varphi-\phi|\rangle}}_{<\sqrt{\epsilon^{2}}\hbox{
from \eqref{Eq:BoundIntVarphi-phiCMEpsilonRdRd}}}$ $\displaystyle\leq
2\sqrt{|C_{M}|(\mathbb{R}^{d}\times\mathbb{R}^{d})}\|\varphi\|_{\infty}\epsilon,$

On the other hand, with similar arguments one has for $(d)$:

$\displaystyle(d)$ $\displaystyle=\left|\langle
C_{M},\varphi\otimes(\varphi-\phi)\rangle+\langle
C_{M},(\varphi-\phi)\otimes\phi\rangle\right|$ (81)
$\displaystyle\leq\sqrt{\langle C_{M},\varphi\otimes\varphi\rangle\langle
C_{M},(\varphi-\phi)\otimes(\varphi-\phi)\rangle}+\sqrt{\langle
C_{M},\phi\otimes\phi\rangle\langle
C_{M},(\varphi-\phi)\otimes(\varphi-\phi)\rangle}$ $\displaystyle\leq
2\sqrt{|C_{M}|(\mathbb{R}^{d}\times\mathbb{R}^{d})}\|\varphi\|_{\infty}\epsilon.$

These bounds for $(a),(b),(c)$ and $(d)$ allow to conclude

$\left|\Lambda(\varphi,\varphi)-\langle
C_{M},\varphi\otimes\varphi\rangle\right|\leq(2+4\sqrt{|C_{M}|(\mathbb{R}^{d}\times\mathbb{R}^{d})}\|\varphi\|_{\infty})\epsilon.$
(82)

Since $\epsilon$ was arbitrary, this implies $\Lambda(\varphi,\varphi)=\langle
C_{M},\varphi\otimes\varphi\rangle$ for every $\varphi$ of the form
$\varphi=\mathbf{1}_{I}$. By bi-linearity of $\Lambda$, we can easily extend
this result to every $\varphi$ in the space

$\mathcal{E}:=\operatorname{span}\\{\ \mathbf{1}_{I}\ \big{|}\
I\subset\mathbb{R}^{d}\hbox{ rectangle }\\}.$ (83)

Now, in order to extend this result to any
$\varphi\in\mathcal{M}_{B}(\mathbb{R}^{d})$, we use Lusin’s Theorem 3.1
applied to the space $\mathbb{R}^{d}$ with the finite measure
$|C_{M}|(\cdot\times\mathbb{R}^{d})$. Hence, given the measurable function
$\varphi$ and given $\epsilon>0$, there exists a closed set
$E\subset\mathbb{R}^{d}$ such that $\varphi$ is continuous over $E$ and such
that $|C_{M}|(E^{c}\times\mathbb{R}^{d})<\epsilon^{2}$. In order to
approximate $\varphi$ over $E$, we can consider a typical Riemann-alike
approximation by the functions

$\phi_{n}:=\sum_{j=1}^{n}\varphi(x_{j}^{n})\mathbf{1}_{I_{j}^{n}},$ (84)

where for each $n$, $(I_{j}^{n})_{j=1,...,n}$ is a finite collection of
rectangles forming a partition of a subset $K_{n}$ of $\mathbb{R}^{d}$,
satisfying that $K_{n}\nearrow\mathbb{R}^{d}$ and
$\displaystyle\max_{j=1,...,n}\operatorname{diam}(I_{j}^{n})\to 0$ as
$n\to\infty$, and $x_{j}^{n}\in I_{j}^{n}$ is an arbitrary tag-point. With
those conditions it is not difficult to prove that
$\phi_{n}(x)\xrightarrow[n\to\infty]{}\varphi(x)$ for every $x$ at which
$\varphi$ is continuous, hence in particular we have $\phi_{n}\to\varphi$
point-wise over $E$. In addition, one has
$\|\phi_{n}\|_{\infty}\leq\|\varphi\|_{\infty}$ for every $n$, and of course
$\phi_{n}\in\mathcal{E}$. Thus, from LDCT we can always find a function
$\phi\in\mathcal{E}$ with $\|\phi\|_{\infty}\leq\|\varphi\|_{\infty}$ such
that

$\int_{E\times E}|\varphi-\phi|\otimes|\varphi-\phi|d|C_{M}|<\epsilon^{2}.$
(85)

Using $\||\varphi-\phi|\otimes|\varphi-\phi|\|_{\infty}\leq
4\|\varphi\|^{2}_{\infty}$ and the symmetry of $|C_{M}|$ we obtain

$\displaystyle\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|\varphi-\phi|\otimes|\varphi-\phi|d|C_{M}|$
$\displaystyle=\int_{E\times
E}|\varphi-\phi|\otimes|\varphi-\phi|d|C_{M}|+\int_{E\times
E^{c}}|\varphi-\phi|\otimes|\varphi-\phi|d|C_{M}|$ (86)
$\displaystyle+\int_{E^{c}\times
E}|\varphi-\phi|\otimes|\varphi-\phi|d|C_{M}|+\int_{E^{c}\times
E^{c}}|\varphi-\phi|\otimes|\varphi-\phi|d|C_{M}|$
$\displaystyle\leq\epsilon^{2}+8\|\varphi\|^{2}_{\infty}|C_{M}|(E^{c}\times
E)+4\|\varphi\|^{2}_{\infty}|C_{M}|(E^{c}\times E^{c})$
$\displaystyle\leq\epsilon^{2}+12\|\varphi\|^{2}_{\infty}|C_{M}|(E^{c}\times\mathbb{R}^{d})$
$\displaystyle\leq(1+12\|\varphi\|_{\infty}^{2})\epsilon^{2}.$

With this setted up, we can study the expression
$|\Lambda(\varphi,\varphi)-\langle C_{M},\varphi\otimes\varphi\rangle|$ for
any $\varphi\in\mathcal{M}_{B}(\mathbb{R}^{d})$ by using the same arguments
exposed in (78), using $\phi\in\mathcal{E}$ constructed as above. Expressions
$(a)$ and $(c)$ can be bounded by $\epsilon$ for some $n$ by similar
arguments. For expression $(b)$ we can follow arguments (80) and use the
result (86) and obtain

$\left|\sum_{j\leq
n}\sigma_{X_{j}}^{2}|\langle\mu_{j},\varphi\rangle|^{2}-\sum_{j\leq
n}\sigma_{X_{j}}^{2}|\langle\mu_{j},\phi\rangle|^{2}\right|\leq
2\sqrt{|C_{M}|(\mathbb{R}^{d}\times\mathbb{R}^{d})}\|\varphi\|_{\infty}\sqrt{(1+12\|\varphi\|^{2}_{\infty})}\epsilon.$
(87)

The same arguments are given to bound $(d)$ as well, starting from (81):

$|\langle C_{M},\varphi\otimes\varphi\rangle-\langle
C_{M},\phi\otimes\phi\rangle|\leq
2\sqrt{|C_{M}|(\mathbb{R}^{d}\times\mathbb{R}^{d})}\|\varphi\|_{\infty}\sqrt{(1+12\|\varphi\|^{2}_{\infty})}\epsilon.$
(88)

We can thus conclude

$\left|\Lambda(\varphi,\varphi)-\langle
C_{M},\varphi\otimes\varphi\rangle\right|\leq\big{(}2+4\sqrt{|C_{M}|(\mathbb{R}^{d}\times\mathbb{R}^{d})}\|\varphi\|_{\infty}\sqrt{(1+12\|\varphi\|^{2}_{\infty})}\big{)}\epsilon.$
(89)

Since $\varphi$ and $\epsilon$ were arbitrary, we finally obtain

$\Lambda(\varphi,\varphi)=\langle
C_{M},\varphi\otimes\varphi\rangle=\sum_{j\in\mathbb{N}}\sigma_{X_{j}}^{2}|\langle\mu_{j},\varphi\rangle|^{2},\quad\forall\varphi\in\mathcal{M}_{B}(\mathbb{R}^{d}).$
(90)

This proves hence the result (38).

Concerning the expansion of the covariance $C_{M}$, it is clear that (39)
holds in the sense

$\langle
C_{M},\varphi\otimes\phi\rangle=\sum_{j\in\mathbb{N}}\sigma_{X_{j}}^{2}\langle\mu_{j},\varphi\rangle\langle\mu_{j},\phi\rangle,\quad\forall\varphi,\phi\in\mathcal{M}_{B}(\mathbb{R}^{d}).$
(91)

Let us prove the proposed slightly stronger mode of convergence. Let
$\varphi\in\mathcal{M}_{B}(\mathbb{R}^{d})$. We use expression (18) for the
total-variation measure of $\langle C_{M-M_{n}},\varphi\otimes\cdot\rangle$.
We remind expression (70) which implies both $C_{M-M_{n}}=C_{M}-C_{M_{n}}$ and
that $\langle C_{M-M_{n}},\phi\otimes\phi\rangle\leq\langle
C_{M},\phi\otimes\phi\rangle$ for every
$\phi\in\mathcal{M}_{B}(\mathbb{R}^{d})$. We have hence

$\displaystyle\left|\langle
C_{M}-C_{M_{n}},\varphi\otimes\cdot\rangle\right|(\mathbb{R}^{d})$
$\displaystyle=\left|\langle
C_{M-M_{n}},\varphi\otimes\cdot\rangle\right|(\mathbb{R}^{d})$ (92)
$\displaystyle=\sup_{\phi\in\mathcal{M}(\mathbb{R}^{d}),|\phi|=1}\left|\langle
C_{M-M_{n}},\varphi\otimes\phi\rangle\right|$
$\displaystyle\leq\sup_{\phi\in\mathcal{M}(\mathbb{R}^{d}),|\phi|=1}\sqrt{\langle
C_{M-M_{n}},\varphi\otimes\varphi\rangle}\sqrt{\langle
C_{M-M_{n}},\phi\otimes\phi\rangle}$ $\displaystyle\leq\sqrt{\langle
C_{M-M_{n}},\varphi\otimes\varphi\rangle}\sup_{\phi\in\mathcal{M}(\mathbb{R}^{d}),|\phi|=1}\sqrt{\langle
C_{M},\phi\otimes\phi\rangle}$ $\displaystyle\leq\sqrt{\langle
C_{M-M_{n}},\varphi\otimes\varphi\rangle}\sqrt{|C_{M}|(\mathbb{R}^{d}\times\mathbb{R}^{d})}\xrightarrow[n\to\infty]{}0,$

which proves the desired result. $\blacksquare$

For the case of non-finite regular random measures we will use the following
definition.

######  Definition 4.1.

A regular random measure $M$ is said to be regulated by a function, if there
exists a strictly positive and locally bounded function
$f\in\mathcal{M}(\mathbb{R}^{d})$ such that $\frac{1}{f\otimes f}|C_{M}|$ is a
finite measure.

It is not difficult to prove that if $M$ is regulated by $f$, then
$\frac{1}{f}M$ is finite. The following result is obtained immediately from
Theorem 4.1.

######  Theorem 4.2.

let $M$ be a regular random measure over $\mathbb{R}^{d}$ regulated by a
function $f$. Then, there exist a sequence of pairwise uncorrelated random
variables with summable variances $(X_{n})_{n\in\mathbb{N}}$, and a sequence
of real measures over $\mathbb{R}^{d}$, $(\mu_{n})_{n\in\mathbb{N}}$, all of
them such that $\frac{1}{f}|\mu_{n}|$ is a finite measure, such that

$\langle
M,\varphi\rangle=\sum_{j\in\mathbb{N}}X_{j}\langle\mu_{j},\varphi\rangle,\quad\forall\varphi\in\mathcal{M}_{B,c}(\mathbb{R}^{d}),$
(93)

with the series being considered in a mean-square sense. For $C_{M}$ expansion
(39) holds, the convergence of the series being in a separately
$\mathcal{M}_{B,c}(\mathbb{R}^{d})^{*}$-weak${}^{*}-$total-variation-on-
compacts sense, that is,

$\left|\langle C_{M},\varphi\otimes(\cdot)\rangle-\sum_{j\leq
n}\sigma_{X_{j}}^{2}\langle\mu_{j},\varphi\rangle\mu_{j}\right|(K)\xrightarrow[n\to\infty]{}0,\quad\forall\varphi\in\mathcal{M}_{B,c}(\mathbb{R}^{d})\forall
K\subset\mathbb{R}^{d}\hbox{ compact}.$ (94)

Proof: We apply Theorem 4.1 to the finite random measure $\frac{1}{f}M$,
obtaining thus

$\langle\frac{1}{f}M,\phi\rangle=\sum_{j\in\mathbb{N}}X_{j}\langle\nu_{j},\phi\rangle,\quad\phi\in\mathcal{M}_{B}(\mathbb{R}^{d}).$
(95)

The measures $\nu_{j}$ are of course finite. If
$\varphi\in\mathcal{M}_{B,c}(\mathbb{R}^{d})$, since $f$ is locally bounded
then
$f\varphi\in\mathcal{M}_{B,c}(\mathbb{R}^{d})\subset\mathcal{M}_{B}(\mathbb{R}^{d})$.
Since $f$ is positive we have of course

$\langle M,\varphi\rangle=\langle\frac{1}{f}M,f\varphi\rangle,$ (96)

hence

$\langle M,\varphi\rangle=\sum_{j\in\mathbb{N}}X_{j}\langle
f\nu_{j},\varphi\rangle,\quad\varphi\in\mathcal{M}_{B,c}(\mathbb{R}^{d}).$
(97)

Setting $\mu_{j}:=f\nu_{j}$ we obtain then the desired sequence of measures
for which $\frac{1}{f}\mu_{j}=\nu_{j}$ is finite for every $j$. The
convergence of the covariance (94) is obtained following the same arguments
used at the end of the proof of Theorem 4.1. $\blacksquare$

While it is not clear if a general regular random measure $M$ can be regulated
by a function $f$, it is possible to do so for some important classes of
random measures. For instance, tempered random measures which are regulated by
a polynomial. Such random measures determine tempered random distributions
over $\mathbb{R}^{d}$. This is the case for instance of White Noise and of
second-order stationary random measures (that is, with translation invariant
covariance structure). Another important class of random measures when Theorem
4.2 can be applied is the case of orthogonal random measures. An orthogonal
random measure is a regular random measure $M$ such that $C_{M}(A\times
B)=\mu(A\cap B)$ for some positive control measure $\mu$. Since for every
positive (locally-finite) measure $\mu$ we can find a function $f$ such that
$f^{2}$ regulates $\mu$ (in an analogous sense to definition 4.1), we can thus
find $f$ such that $\frac{1}{f}M$ is a finite random measure and thus obtain
the result.

##  5 Concluding remarks

We finish with some remarks about the obtained results.

######  Remark 5.1.

The reader may wonder about a direct connection between the expansion in
Theorem 4.1 and the Hilbert space approach presented in Section 2. For that,
we remark that $\mathcal{O}(\mu)$ belongs to $L^{2}(\mathbb{R}^{d},\nu)$ for
every finite measure $\mu$ ($\nu$ defined as in (42)). If $E$ is the space of
finite measures over $\mathbb{R}^{d}$, we can endow $E$ with the bi-linear
positive-semidefinite form

$(\mu_{1},\mu_{2})_{E}\mapsto\left(\mathcal{O}(\mu_{1}),\mathcal{O}(\mu_{2})\right)_{L^{2}(\mathbb{R}^{d},\nu)}.$
(98)

From the continuity of $\mathcal{O}(\mu)$ and the right-continuity at every
component of $\vec{u}\mapsto\mu(\left(-\infty,\vec{u}\right])$, one can
conclude that $(\mu,\mu)_{E}=0$ if and only if $\mu=0$. The completion of $E$
with this product will be hence an abstract Hilbert space with respect to
which a Karhunen-Loève decomposition as expressed in Section 2 can be stated.

######  Remark 5.2.

Results 4.1 and 4.2 can be used to obtain, as corollaries, diverse forms of
Karhunen-Loève expansions of not regular stochastic processes over
$\mathbb{R}^{d}$. For instance, if we have a process of the form
$Z(\vec{x})=M((-\infty,\vec{x}])$, with $M$ being a finite regular random
measure, then an immediate application of Theorem 4.1 to the measurable and
bounded function $\mathbf{1}_{(-\infty,\vec{x}]}$ implies a traceable
Karhunen-Loève decomposition for $Z$ of the form

$Z=\sum_{n\in\mathbb{N}}X_{n}g_{n},$ (99)

with the functions $g_{n}$ being all bounded an right-continuous at every
component ($g_{n}(\vec{x})=\mu_{j}((-\infty,\vec{x}])$). The convergence of
the series (99) is mean-square-point-wise. This can be applied to the case of
right-continuous with left-limits process, as for example Lévy processes, in
dimension $d=1$. Other examples can be given by choosing parameter depending
functions $\varphi$ in equation (38).

######  Remark 5.3.

For a general non-finite regular random measure $M$ a Karhunen-Loève
decomposition can be done locally. That is, for every bounded Borel set $D$,
$M$ has a decomposition of the form (38) for
$\varphi\in\mathcal{M}_{B}(\mathbb{R}^{d})$ null outside $D$. This because the
restriction measure of $M$ over $D$ is a finite measure for which Theorem 4.1
holds. In such case the measures $\mu_{j}$ and the random variables $X_{j}$
depend upon the set $D$.

######  Remark 5.4.

Of course, if the process $M$ is Gaussian, the random variables in the
Karhunen-Loève expansion are independent. It is also possible to prove in that
case, that the convergence of the series (38) holds almost-surely.

##  Appendix A Proof of Stochastic Fubini Theorem

The first issue with Stochastic Fubini theorem 3.2 is the proper definition of
the iterated integrals presented in (26). From condition (i), a typical use of
deterministic Fubini Theorem implies that the function
$\int_{\mathbb{R}^{m}}\psi(\cdot,u)d\mu(u)\otimes\int_{\mathbb{R}^{m}}\psi(\cdot,v)d\mu(v)$
is in $L^{1}(\mathbb{R}^{d}\times\mathbb{R}^{d},|C_{M}|)$. Hence, the integral
of $\int_{\mathbb{R}^{m}}\psi(\cdot,u)d\mu(u)$ with respect to $M$ over
$\mathbb{R}^{d}$, which is the integral at the left side of (26) is a well-
defined stochastic integral. Let us look at the one at the right side.

As it is mentioned in the theorem, from condition (i) it follows that the
function

$C_{Z}(u,v):=\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\psi(x,u)\psi(y,v)dC_{M}(x,y)$
(100)

is $|\mu|\otimes|\mu|$-almost-everywhere defined over
$\mathbb{R}^{m}\times\mathbb{R}^{m}$. It follows that the stochastic process
over $\mathbb{R}^{m}$

$Z(u)=\int_{\mathbb{R}^{d}}\psi(x,u)dM(x),$ (101)

is well-defined for every $u\in\mathbb{R}^{m}$ outside à $|\mu|$-null set $D$,
and its covariance function is $C_{Z}$. The extra continuity supposition (25)
says that $C_{Z}$ is continuous over $E\times E$, which implies that $Z$ is
mean-square continuous over $E$ (we may assume $D\subset E^{c}$). In such
case, the definition of the stochastic integral

$\int_{\mathbb{R}^{m}}Z(u)d\mu(u)$ (102)

can be done with a Riemann approximation which we will make precise. We
consider for each $n\in\mathbb{N}$ a finite collection of rectangles
$(I_{j}^{n})_{j=1,...,n}$ forming a partition of a subset $K_{n}$ such that
$K_{n}\nearrow\mathbb{R}^{m}$ and
$\displaystyle\max_{j=1,...,n}\operatorname{diam}(I_{j}^{n})\to 0$ as
$n\to\infty$. For every $I_{j}^{n}$ such that $E\cap I_{j}^{n}\neq\emptyset$,
we set an arbitrary tag-point $u_{j}^{n}\in E\cap I_{j}^{n}$. Then, we have

$\int_{\mathbb{R}^{m}}Z(u)d\mu(u):=\lim_{n\to\infty}\sum_{j=1}^{n}Z(u_{j}^{n})\mu(I_{j}^{n}\cap
E),$ (103)

where the limit is considered in a mean-square sense. It is known that such
limit exists and it is not depending on the selection of the partitions or
tag-points when (Soong TT, 1973, Theorem 4.5.2)

$\int_{\mathbb{R}^{m}\times\mathbb{R}^{m}}|C_{Z}|d|\mu|\otimes|\mu|<\infty.$
(104)

This is guaranteed by condition (i). Hence the integral (102) is well-defined,
and it coincides with the integral at the right-side of equation (26)222The
author is not aware of any result concerning the proper definition of the
mean-square stochastic integral (102) when $Z$ has a measurable covariance
$C_{Z}$ with no any other regularity property more than (104). This is the
only reason why the extra continuity condition (25) has been required in the
Stochastic Fubini Theorem 3.2. It is expected that less strict conditions may
also work. (Gill, 1987) proposes, for instance, sufficient second-order
conditions on $Z$ which make it have a measurable modification, which could be
useful in this aim.. Of course, we have exploited the fact that
$|\mu|(E^{c})=0$ in order to implicitly have

$\int_{\mathbb{R}^{m}}Z(u)d\mu(u)=\int_{E}Z(u)d\mu(u)\quad,\quad\int_{E^{c}}Z(u)d\mu(u)=0.$
(105)

Let us now consider the variance of the difference between the integrals

$a:=\mathbb{E}\left(\left|\int_{\mathbb{R}^{m}}\int_{\mathbb{R}^{d}}\psi(x,u)dM(x)d\mu(u)-\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{m}}\psi(x,u)d\mu(u)dM(x)\right|^{2}\right).$
(106)

Using the limit expression (103) one has

$\displaystyle a=$
$\displaystyle\lim_{n\to\infty}\mathbb{E}\left(\left|\sum_{j=1}^{n}\int_{\mathbb{R}^{d}}\psi(x,u_{j}^{n})dM(x)\mu(I_{j}^{n}\cap
E)-\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{m}}\psi(y,v)d\mu(v)dM(y)\right|^{2}\right)$
(107) $\displaystyle=$
$\displaystyle\lim_{n\to\infty}\sum_{j=1}^{n}\sum_{k=1}^{n}\mathbb{E}\left(\int_{\mathbb{R}^{d}}\psi(x,u_{j}^{n})dM(x)\int_{\mathbb{R}^{d}}\psi(y,u_{k}^{n})dM(x)\right)\mu(I_{j}^{n}\cap
E)\mu(I_{k}^{n}\cap E)$
$\displaystyle-\lim_{n\to\infty}2\sum_{j=1}^{n}\mathbb{E}\left(\int_{\mathbb{R}^{d}}\psi(x,u_{j}^{n})dM(x)\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{m}}\psi(y,v)d\mu(v)dM(y)\right)\mu(I_{j}^{n}\cap
E)$
$\displaystyle+\mathbb{E}\left(\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{m}}\psi(x,u)d\mu(u)dM(x)\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{m}}\psi(y,v)d\mu(v)dM(y)\right).$

For the expectations we use (24) in order to obtain

$\displaystyle a=$
$\displaystyle\lim_{n\to\infty}\sum_{j=1}^{n}\sum_{k=1}^{n}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\psi(x,u_{j}^{n})\psi(y,u_{k}^{n})dC_{M}(x,y)\mu(I_{j}^{n}\cap
E)\mu(I_{k}^{n}\cap E)$ (108)
$\displaystyle-2\lim_{n\to\infty}\sum_{j=1}^{n}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\psi(x,u_{j}^{n})\int_{\mathbb{R}^{m}}\psi(y,v)d\mu(v)dC_{M}(x,y)\mu(I_{j}^{n}\cap
E)$
$\displaystyle+\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\int_{\mathbb{R}^{m}}\psi(x,u)d\mu(u)\int_{\mathbb{R}^{m}}\psi(y,v)d\mu(v)dC_{M}(x,y).$

Using the continuity of the function $C_{Z}$ over the $|\mu|\otimes|\mu|$-full
set $E\times E$, we have the Riemann approximation of the integral

$\displaystyle\lim_{n\to\infty}\sum_{j=1}^{n}\sum_{k=1}^{n}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\psi(x,u_{j}^{n})$
$\displaystyle\psi(y,u_{k}^{n})dC_{M}(x,y)\mu(I_{j}^{n}\cap
E)\mu(I_{k}^{n}\cap E)$ (109)
$\displaystyle=\int_{\mathbb{R}^{m}\times\mathbb{R}^{m}}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\psi(x,u)\psi(y,v)dC_{M}(x,y)d(\mu\otimes\mu)(u,v),$

which exists due to condition (i). On the other hand, by deterministic Fubini
Theorem,

$\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\psi(x,u)\int_{\mathbb{R}^{m}}\psi(y,v)d\mu(v)dC_{M}(x,y)=\int_{\mathbb{R}^{m}}C_{Z}(u,v)d\mu(v),$
(110)

$|\mu|(u)$-almost everywhere. Let us verify that the function
$u\mapsto\int_{\mathbb{R}^{m}}C_{Z}(u,v)d\mu(v)$ is continuous over $E$. For
that, we see

$\displaystyle\left|\int_{\mathbb{R}^{m}}C_{Z}(u,v)-C_{Z}(w,v)d\mu(v)\right|$
$\displaystyle=\left|\int_{E}C_{Z}(u,v)-C_{Z}(w,v)d\mu(v)\right|$ (111)
$\displaystyle\leq\int_{E}|C_{Z}(u,v)-C_{Z}(w,v)|d|\mu|(v).$

Since $|\mu|$ is finite and $C_{Z}$ is continuous over $E\times E$, it is
clear that (111) goes to $0$ as $w\to v\in E$, proving hence the continuity
over $E$ of $u\mapsto\int_{\mathbb{R}^{m}}C_{Z}(u,v)d\mu(v)$. From (i) this
function is integrable with respect to $\mu$ and its integral can be
approximated by Riemann sums. Hence,

$\displaystyle\lim_{n\to\infty}\sum_{j=1}^{n}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\psi(x,u_{j}^{n})$
$\displaystyle\int_{\mathbb{R}^{m}}\psi(y,v)d\mu(v)dC_{M}(x,y)\mu(I_{j}^{n}\cap
E)$ (112)
$\displaystyle=\int_{\mathbb{R}^{m}}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\psi(x,u)\int_{\mathbb{R}^{m}}\psi(y,v)d\mu(v)dC_{M}(x,y)d\mu(u).$

Using thus the limits (109) and (112), one obtains

$\displaystyle a=$
$\displaystyle\int_{\mathbb{R}^{m}\times\mathbb{R}^{m}}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\psi(x,u)\psi(y,v)dC_{M}(x,y)d(\mu\otimes\mu)(u,v)$
(113)
$\displaystyle-2\int_{\mathbb{R}^{m}}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\psi(x,u)\int_{\mathbb{R}^{m}}\psi(y,v)d\mu(v)dC_{M}(x,y)d\mu(u)$
$\displaystyle+\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\int_{\mathbb{R}^{m}}\psi(x,u)d\mu(u)\int_{\mathbb{R}^{m}}\psi(y,v)d\mu(v)dC_{M}(x,y).$

But the deterministic Fubini Theorem, guaranteed by condition (i) and the fact
that $C_{M}$ is a measure, allows to conclude the equality between all these
deterministic multiple integrals, implying $a=0$. This proves the equality of
the stochastic integrals (26) as random variables in
$L^{2}(\Omega,\mathcal{A},\mathbb{P})$. $\blacksquare$

##  Appendix B Properties of the Regularising Operator

The first claim in Section 3.3 concerning the operator $\mathcal{O}$ acting
over a finite measure $\mu$ is that $\mathcal{O}(\mu)$ is a continuous
function. This holds because it consists of $d$ iterated Riemann integrals of
right-continuous bounded functions. The derivative property (31) is justified
by a convolution argument which we make precise for the case $d=1$. In such
case, we may re-write $\mathcal{O}(\mu)$ as

$\mathcal{O}(\mu)=\mathbf{1}_{[0,\infty)}\ast\left[\left(\mathbf{1}_{[0,\infty)}\ast\mu\right)\mathbf{1}_{[0,\infty)}\right]-\mathbf{1}_{(-\infty,0)}\ast\left[\left(\mathbf{1}_{[0,\infty)}\ast\mu\right)\mathbf{1}_{(-\infty,0)}\right].$
(114)

where $\ast$ denotes the convolution operation. Using the properties of the
convolution with respect to derivatives, that the Dirac measure $\delta$ (at
$0$) is its identity element, and that
$\frac{d}{dx}\mathbf{1}_{[0,\infty)}=\delta$, one has

$\displaystyle\frac{d}{dx}\mathcal{O}(\mu)$
$\displaystyle=\delta\ast\left[\left(\mathbf{1}_{[0,\infty)}\ast\mu\right)\mathbf{1}_{[0,\infty)}\right]-(-\delta)\ast\left[\left(\mathbf{1}_{[0,\infty)}\ast\mu\right)\mathbf{1}_{(-\infty,0)}\right]$
(115)
$\displaystyle=\left(\mathbf{1}_{[0,\infty)}\ast\mu\right)\mathbf{1}_{[0,\infty)}+\left(\mathbf{1}_{[0,\infty)}\ast\mu\right)\mathbf{1}_{(-\infty,0)}$
$\displaystyle=\mathbf{1}_{[0,\infty)}\ast\mu.$

Deriving again,

$\frac{d^{2}}{dx^{2}}\mathcal{O}(\mu)=\delta\ast\mu=\mu.$ (116)

The case for $d>1$ is analogous though requiring a more tedious notation.

When applying operator $\mathcal{O}$ to a regular finite random measure $M$,
the question about the proper definition of the process $\mathcal{O}(M)$
arises. Let us verify its definition. First, the process

$V(\vec{u}):=M((-\infty,\vec{u}]),\quad\vec{u}\in\mathbb{R}^{d},$ (117)

has a covariance function (Eq. (24))

$C_{V}(\vec{u},\vec{v})=C_{M}((-\infty,\vec{u}]\times(-\infty,\vec{v}]).$
(118)

Consider the function
$\vec{u}\mapsto|C_{M}|(\mathbb{R}^{d}\times(-\infty,\vec{u}])$. This function
is increasing and bounded (and càdlàg) in each component of $\vec{u}$ when the
other are fixed. It is known that such kind of functions are continuous
almost-everywhere over $\mathbb{R}^{d}$. Let $E$ be the set of continuity
points of such function. Let us verify that $C_{V}$ is continuous over
$E\times E$. Since $C_{V}$ is a covariance function, it is continuous at
$E\times E$ if it is continuous at diagonal points $(\vec{u},\vec{u})\in
E\times E$ (Soong TT, 1973, Theorem 4.3.3). For such points we have

$\displaystyle|C_{V}(\vec{u}+\vec{h},\vec{u}+\vec{k})-C_{V}(\vec{u},\vec{u})|\leq$
$\displaystyle|C_{V}(\vec{u}+\vec{h},\vec{u}+\vec{k})-C_{V}(\vec{u}+\vec{h},\vec{u})|$
(119) $\displaystyle+|C_{V}(\vec{u}+\vec{h},\vec{u})-C_{V}(\vec{u},\vec{u})|$

Now, using (118), we see that differences of the form
$|C_{V}(\vec{a},\vec{b})-C_{V}(\vec{a},\vec{c})|$ are actually of the form
$|C_{M}(A\times B)-C_{M}(A\times C)|$ for three sets $A,B,C$. Since $C_{M}$ is
a finite measure, one has

$|C_{M}(A\times B)-C_{M}(A\times C)|\leq|C_{M}|(A\times(B\bigtriangleup
C))\leq|C_{M}|(\mathbb{R}^{d}\times(B\bigtriangleup C)),$ (120)

where $B\bigtriangleup C$ denotes the symmetric difference between the sets
$B$ and $C$. In our case, the sets which play the role of $B$ and $C$ are of
the form $(-\infty,\vec{u}+\vec{h}]$ and $(-\infty,\vec{u}]$. In such case we
have

$(-\infty,\vec{u}+\vec{h}]\bigtriangleup(-\infty,\vec{u}]\subset\overline{B_{|\vec{h}|_{\infty}}^{\infty}(\vec{u})},$
(121)

where $\overline{B_{|\vec{h}|}^{\infty}(\vec{u})}$ denotes the closed ball of
radius $|\vec{h}|_{\infty}$ centred at $\vec{u}$ with respect to the
$\infty-$norm over $\mathbb{R}^{d}$. $|\vec{h}|_{\infty}$ denotes the
$\infty-$norm of the vector $\vec{h}\in\mathbb{R}^{d}$. We conclude thus

$|C_{V}(\vec{u}+\vec{h},\vec{u}+\vec{k})-C_{V}(\vec{u},\vec{u})|\leq|C_{M}|(\mathbb{R}^{d}\times\overline{B_{|\vec{k}|}^{\infty}(\vec{u})})+|C_{M}|(\mathbb{R}^{d}\times\overline{B_{|\vec{h}|}^{\infty}(\vec{u})})$
(122)

(we have used the symmetry of $C_{M}$). Since the function
$\vec{u}\mapsto|C_{M}|(\mathbb{R}^{d}\times(-\infty,\vec{u}])$ is continuous
at $\vec{u}\in E$, expression (122) goes to $0$ as $\vec{h},\vec{k}\to 0$,
thus $C_{V}$ is continuous over $E\times E$.

The definition of $\mathcal{O}(M)$ involves hence the stochastic integral of a
stochastic process with respect to the Lebesgue measure (a Riemann integral
actually), the process being continuous almost-everywhere. This integral is
hence well-defined, as it has been stated in Appendix A. The expression of the
covariance (34) follows from the properties of such integrals.

The derivative relation (36) comes from Stochastic Fubini Theorem. Indeed, if
$\varphi\in\mathscr{D}(\mathbb{R}^{d})$ then,

$\displaystyle\int_{\mathbb{R}^{d}}\mathcal{O}(M)(\vec{x})\frac{\partial^{2d}\varphi}{\partial
x_{1}^{2}...\partial x_{d}^{2}}(\vec{x})d\vec{x}$
$\displaystyle=\int_{\mathbb{R}^{d}}\int_{0}^{\vec{x}}\int_{\mathbb{R}^{d}}\mathbf{1}_{(-\infty,\vec{u}]}(s)dM(s)d\vec{u}\frac{\partial^{2d}\varphi}{\partial
x_{1}^{2}...\partial x_{d}^{2}}(\vec{x})d\vec{x}$ (123)
$\displaystyle=\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\int_{0}^{\vec{x}}\delta_{s}((-\infty,\vec{u}])d\vec{u}\frac{\partial^{2d}\varphi}{\partial
x_{1}^{2}...\partial x_{d}^{2}}(\vec{x})d\vec{x}dM(s)$
$\displaystyle=\int_{\mathbb{R}^{d}}\langle\mathcal{O}(\delta_{s}),\frac{\partial^{2d}\varphi}{\partial
x_{1}^{2}...\partial x_{d}^{2}}\rangle dM(s)$
$\displaystyle=\int_{\mathbb{R}^{d}}\langle\delta_{s},\varphi\rangle
dM(s)=\int_{\mathbb{R}^{d}}\varphi(s)dM(s).$

Here we have used (32). The arguments which verify that the hypotheses in
Stochastic Fubini Theorem 3.2 hold are quite similar to the ones used in the
Proof of Theorem 4.1 (see Eq. (60) and Eq. (62)). The details are left to the
reader.

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