A NOTE ON HERGLOTZ’S THEOREM FOR TIME SERIES ON
FUNCTION SPACES
Anne van Delft and Michael Eichler

arXiv:1801.04262v1 [] 12 Jan 2018

January 15, 2018
Abstract. In this article, we prove Herglotz’s theorem for Hilbert-valued time
series. This requires the notion of an operator-valued measure, which we shall
make precise for our setting. Herglotz’s theorem for functional time series allows
to generalize existing results that are central to frequency domain analysis on
the function space. In particular, we use this result to prove the existence of a
functional Cramér representation of processes with jumps in the spectral distribution, thereby enabling Fourier analysis for processes of which the spectal density
operator is not a well-defined object.
Keywords: Functional data analysis, spectral analysis, time series
2010 Mathematics Subject Classification. Primary: 60G10; Secondary: 62M15.

1. introdu tion
As a result of a surge in data storage techniques, many data sets can be viewed as
being sampled continuously on their domain of definition. It is therefore natural
to think of the data points as being objects and embed them into an appropriate
mathematical space that accounts for the particular properties and structure of the
space. The development of meaningful statistical treatment of these objects is known
as functional data analysis and views each random element as a point in a function
space. Not surprisingly, this has become an active field of research in recent years.
If the random functions can be considered an ordered collection tXt utPZ we call this
collection dependent functional data or a functional time series. The function space
where each Xt takes its values is usually assumed to be the Hilbert space L2 pr0, 1sq,
in which case we can parametrize our functions τ ÞÑ Xt pτ q, τ P r0, 1s.
While the literature on classical time series finds its origin in harmonic analysis, the literature on its functional counterpart started in the time domain. The
frequency domain arises however quite naturally in the analysis of dependent functional data. The second order dependence structure encodes the relevant information
on the shape and smoothness properties of the random curves. It provides a way
to optimally extract the intrinisically infinite variation carried by the random functions to lower dimension. In case the functional time series is weakly stationary, the
second order dependence structure can be specified in the time domain through an
Ruhr-Universität Bochum, Fakultät für Mathematik, 44780 Bochum, Germany
E-mail address: anne.vandelft@rub.de (A. van Delft)
Department of Quantitative Economics, Maastricht University, P.O. Box 616, 6200 MD Maastricht,
The Netherlands
E-mail address: m.eichler@maastrichtuniversity.nl (M. Eichler)
1

2

ANNE VAN DELFT AND MICHAEL EICHLER

infinite sequence of lag h covariance operators
“
‰
Ch “ pX0 ´ mq b pXh ´ mq ,

E

Z

hP ,

where m is the mean function of X, which is the unique element of H such that
xm, gy “

ExX, gy,

g P H.

Unlike independent functional data, where one only needs to consider the within
curves dynamics as captured by the operator C0 , functional time series require to
take into account also all the between curve dynamics as given by the infinite sequence of lag covariance operators Ch for h ‰ 0. The full second order dynamics
are then more straightforwardly captured in the frequency domain, and an initial framework for Fourier analysis of random functions was therefore developed
in Panaretos and Tavakoli (2013b). Their framework of frequency domain-based
inference is however restricted to processes for which the notion of a spectral density operator, defined as the Fourier transform of the sequence of h-lag covariance
operators,
1 ÿ
Fω “
Ch e´ihω ,
ω P r´π, πs,
2π hPZ

is well-defined. In this case, the autocovariance operator at lag h can itself be
represented as
żπ
Ch “
eihω Fω dω,
(1.1)
´π

where the convergence holds in the appropriate operator norm. For processes of
which Fω has absolutely summable eigenvalues, Panaretos and Tavakoli (2013a) derived a functional Cramér representation and showed that the eigenfunctions of Fω
allow an optimal lower dimension representation. In this case, the spectral density
operator can be seen to take the same role as the covariance operator for independent functional data in the classical Karhunen-Loève expansion (Karhunen, 1947;
Loève, 1948). Van Delft and Eichler (2018) extended frequency domain-based inference for functional data to locally stationary processes allowing thus to relax the
notion of weak stationarity and to consider time-dependent second order dynamics
through a time-varying spectral density operator. Since frequency domain based
inference does not require structural modeling assumptions other than weak dependence conditions, it has proved helpful in the construction of stationarity tests (see
e.g., Aue and van Delft, 2017; van Delft et al., 2017).
All of the aforementioned literature is restricted to processes with well-defined
spectral density operators in S1 pHq. This excludes many interesting processes with
highly persistent cyclical or seasonal components. For instance, the supply and demand curves for electricity prices show a strong daily as well as weakly seasonal
pattern. In order to extend frequency domain analysis to such Hilbert-valued processes for which the spectral density operator does not necessarily exist, we establish
functional versions of the two fundamental results that lie at the core of frequency
domain analysis for classical stationary time series: Herglotz’s Theorem and the
Cramér Representation Theorem. The structure of this note is as follows. We start
by introducing the necessary notation and terminology. In section 3, we establish the
existence of a functional Herglotz’s theorem. For this, we make precise the concept
of an operator-valued measure and the notion of operator-valued kernel functions.
In section 4, Herglotz’s theorem is then in turn used to prove a generalized functional
Cramér representation for weakly stationary Hilbert-valued time series.

A NOTE ON HERGLOTZ’S THEOREM FOR TIME SERIES ON FUNCTION SPACES

3

2. Notation and preliminaries
2.1. The function space
We first introduce some necessary notation. Let pT, Bq be a measurable space with
σ-finite measure µ. Furthermore, let E be a Banach space with norm } ¨ }E and
equipped with the Borel σ-algebra. We then define LpE pT, µq as the Banach space of
all strongly measurable functions f : T Ñ E with finite norm
´ż
¯1
p
p
p
}f }LE pT,µq “
}f pτ q}E dµpτ q
for 1 ď p ă 8 and with finite norm
“ inf
}f }L8
E pT,µq

sup }f pτ q}E

µpN q“0 τ PT zN

Lp

for p “ 8. We note that two functions f and g are equal in Lp , denoted as f “ g, if
}f ´ g}LpE pT,µq “ 0. If E is a Hilbert space with inner product x¨, ¨yE then L2E pT, µq
is also a Hilbert space with inner product
ż
xf, gyL2 pT,µq “ xf pτ q, gpτ qyE dµpτ q.
E

For notational convenience, we simply write xf, gy if no ambiguity about the space
L2E pT, µq is possible. In particular, for µ is the Lebesque measure and E “ , we
simply write Lp pT q and denote its norm by } ¨ }p .
We shall extensively make use of linear operators on a Hilbert space H. A
linear operator on a Hilbert space H is a function A : H Ñ H that preserves
the operations of scalar multiplication and addition. We shall denote the class of
bounded linear operators by LpHq and its norm by ~¨~L . Furthermore, the class of
trace-class operators and Hilbert-Schmidt operators will be denoted by S1 pHq and
S2 pHq, respectively and their norms by ~¨~1 and ~¨~2 . Equipped with these norms
pLpHq, ~¨~L q and pS1 pHq, ~¨~1 q form Banach spaces while pS2 pHq, ~¨~2 q forms a
Hilbert space with inner product x¨, ¨yS2 . An operator A P LpHq is called selfadjoint if xAf, gy “ xf, Agy for all f, g P H, while we say it is non-negative definite
if xAg, gy ě 0 for all g P H. It will be convenient to denote the respective operator
subspaces of self-adjoint and non-negative operators with p¨q: and p¨q` , respectively.
It is straightforward to verify that LpHq` Ď LpHq: and Sp pHq` Ď Sp pHq:.

C

2.2. Functional time series

Z

We define a functional time series tXt : t P u as a sequence of random elements
on a probability space pΩ, A, q taking values in H :“ L2 pr0, 1sq, the separable
Hilbert space of square integrable functions. Additionally, we denote H “ L2H pΩ, q
the Hilbert space of all H-valued random variables X with finite second moment
}X}22 ă 8. We shall assume throughout that Xt P H. A functional time series
X “ pXt : t P q is called strictly stationary if, for all finite sets of indices J Ă Z,
the joint distribution of pXt`j : j P Jq does not depend on t P . Similarly, X
is weakly stationary if its first- and second-order moments exist and are invariant
under translation in time. Without loss of generality, we shall assume the mean
function is zero, i.e., m “ 0 P H. In this case, the h-th lag covariance operators Ch
can be defined by
“
‰
xCh g1 , g2 y “ x pXh b X0 qg1 , g2 y “ xg1 , X0 y xXh , g2 y ,
g1 , g2 P H,

P

E

P

Z

Z

E

E

4

ANNE VAN DELFT AND MICHAEL EICHLER

Z

and belongs to S2 pHq for all h P . Parseval’s identity moreover implies that
C0 belongs to S1 pHq. In the setting of time series analysis, it is however natural to
assume that the strength of serial correlation decreases as the time distance between
the random functions increases and hence that at least tCh uhPZ P S1 pHq. This will
be assumed in section 4.
In the literature, frequency domain analysis for weakly stationary functional time
series so far has been restricted to processes of which the dependence structure
decays at a much faster rate, namely
ÿ
~Ch ~1 ă 8.
(2.1)

Z

hP

Under this condition, it is straightforward to show that the autocovariance operator
Ch forms a Fourier pair with the spectral density operator given by
1 ÿ
Fω “
Ch e´ihω ,
2π hPZ
where the convergence holds in ~¨~1 and the spectral density operator acts on H.
Given assumption (2.1), a classical Cesàro-sum argument (see e.g. Brillinger, 1981),
was used by Panaretos and Tavakoli (2013b) to derive that Fω is non-negative definite and hence belongs to S1 pHq` . As already mentioned in the introduction, the
autocovariance operator at lag h itself can then be represented as
żπ
Ch “
eihω Fω dω,
(2.2)
´π

where the convergence holds in ~¨~1 . Panaretos and Tavakoli (2013a) showed that
a zero mean stationary functional time series X satisfying condition (2.1) admits a
functional spectral representation of the form
żπ
Xt “
eiωt dZω
a.s.,
(2.3)
´π

where Zω is a functional orthogonal increment process
such that, for fixed ω, Zω is
şω
2
2
a random element in LC pr0, 1sq with }Zω }2 “ ´π ~Fλ ~1 dλ. If the summability
conditions as in (2.1) do not hold, the spectral density operator does not necessarily
exist and as a consequence (2.2) and (2.3) are no longer valid. In the next section,
we shall derive a frequency domain representation of tCh u without imposing any
assumptions on its rate of decay. In section 4, we use this representation to derive
a functional Cramer representation that can be seen as a true generalization of the
classisal Cramér representation theorem to the function space.

E

3. Herglotz's Theorem on a fun tion spa e
In this section we derive a functional generalization of the classical Herglotz’s theorem. Note that it is intuitive from (2.2) that if we do not have a density operator,
then the measure itself must be operator-valued for the equation to be balanced. A
first naive question therefore would be if an operator with representation
żπ
(3.1)
eihω dF pωq,
hP ,
´π

Z

where F is an operator-valued measure on r´π, πs, exists. Secondly, what properties
must an operator possess to be represented by such an integral. From the classical
Herglotz’s theorem, we know that the non-negative definite complex-valued function

A NOTE ON HERGLOTZ’S THEOREM FOR TIME SERIES ON FUNCTION SPACES

5

on the integers are precisely those that can be identified to have a frequency domain
representation with respect to a finite Radon measure.
Theorem 3.1 (Herglotz’s theorem). A function γp¨q :
definite if and only if
żπ
γphq “
eihω dF pωq,
hP

Z Ñ C is non-negative

Z

π

where F p¨q is a right-continuous, non-decreasing bounded function on r´π, πs with
F p´πq “ 0.
Here, the so-called spectral distribution function F p¨q with F p´πq “ 0 is uniquely
determined by the covariance function γphq, h P . To extend this result to our
functional setting, we require the notion of non-negative definiteness of operatorvalued functions on
as well as definitions of operator-valued measures and of
integrals with respect to such measures. For the former, we proceed as in the scalarvalued case by help of non-negative operator-valued kernels.

Z

Z

Definition 3.2.
(i) A function c :
kernel if

Z ˆ Z Ñ LpHq is called a non-negative definite LpHq-valued
n
ÿ

xcpi, jq gj , gi y ě 0

i,j“1

N

for all g1 , . . . , gn P H, and n P .
Ñ LpHq is called non-negative definite if the kernel c :
(ii) A function C :
ˆ Ñ LpHq defined by cpi, jq “ Cpi ´ jq is a non-negative definite kernel.

Z Z

Z

Non-negative definite operator-valued kernels are an extremely powerful tool in
functional analysis, especially in operator theory and representation theory. A generalization of this concept to general involutive semigroups other than
can be
found in Neeb (2000). For Hilbert-valued time series, Definition 3.2 provides a link
between the properties of the covariance kernel and the corresponding covariance operator viewed as a function on the integers. More specifically, we have the following
result.

Z

Z

H

Proposition 3.3. Let tXt : t P u be a weakly stationary process in with secondorder dependence structure given by the sequence of h-lag covariance operators tCh uhPZ .
Then Cp¨q : Ñ LpHq is a non-negative definite function.

Z

Proof. The operator-valued kernel cpi, jq “ Ci´j , i, j P
n
n
ÿ
ÿ
xCpi´jq gj , gi y “
x pXi b Xj qgj , gi y “

E

i,j“1

i,j“1

Z satisfies
n
›ÿ
›
E› xXi, giy›22 ě 0
i“1

and is therefore non-negative definite by Definition 3.2.


3.1. LpHq-valued measures
In this subsection, we explain how LpHq-valued measures can be defined. Let us
first provide some intuition by making an analogy with positive scalar-valued measures. Recall that a positive measure µ on a measurable space pV, Bq is defined as a
countably additive map on B taking values in the compactification r0, 8s of the set

6

ANNE VAN DELFT AND MICHAEL EICHLER

R` :“ r0, 8q.

The compactification is necessary for σ-additivity to hold (although
for finite measures the compact subset r0, µpV qs is sufficient). We note that ` is
an example of a pointed convex cone, that is, it is a convex and nonempty subset
that is closed under nonnegative scalar multiplication and contains the zero
of
element. It is moreover dense in `
8 :“ r0, 8s. Taking this view, a positive measure
more generally can be defined as a countably additive map taking values in the compactification of a pointed convex cone. For the purpose of this paper, we are solely
interested in measures taking values in the compactification of LpHq` , the pointed
convex cone of LpHq consisting of all non-negative elements LpHq. For the general
theory on cone-valued measures we refer to Neeb (2000) and Glockner (2003).
To make such measures meaningful from a practical point of view, it is important
that we can describe them by a set of structure preserving maps that map its dual
(cone) into r0, 8s, i.e., via continuous linear functionals that are `
8 -valued. This
allows to relate an operator-valued measure to a family of positive scalar-valued
measures, where the latter provides us with important information on the operatorvalued measure itself. This is however a non-trivial task. Before we introduce the
definition, we shall heuristically explain the compactification of LpHq` , which we
shall denote by L`
8 . We then explain why this yields a meaningful definition of an
LpHq` -valued measure.

R

R

R

R

3.1.1. Compactification of LpHq`
A natural strategy to obtain a compactification of a cone is to construct a mapping
`
LpHq` Ñ L`
8 . This mapping must be such that each element B P LpHq can be
uniquely identified with an element C P L`
8 . From functional analysis, we know
that we can identify a locally convex space, such as a cone, with its image into its
bidual by means of the canonical injection
x ÞÑ pλ ÞÑ λpxqq,
where x is an element of the cone and λ of its dual cone. To make this mapping
and the dual cone more precise, recall that we can identify the real Banach space
pLpHq: , ~¨~L q with the dual space of pS1 pHq: , ~¨~1 q via the pairing
x¨, ¨y : LpHq: ˆ S1 pHq: Ñ

R,

pB, Aq ÞÑ trpBAq.

(3.2)

The map B ÞÑ φB is an isometric isomorphism from the space of bounded hermitian operators into the dual space of the hermitian trace class operators , i.e.,
S8 pHq: Ñ pS1 pHq: q‹ . For each B P LpHq:, this yields a linear functional on the
space of hermitian trace class operators. However, the functionals pS1 pHq: q‹ are not
necessarily continuous. If we equip LpHq: with the weak operator topology, i.e.,
the coarsest topology that makes all linear functionals of its predual S1 pHq continuous as functions on LpHq: , then the dual pairing provides us with a topological
embedding1
‹‹
φ : LpHq: Ñ pLpHq:q
φpBqpAq “ TrpBAq,
(3.3)
:
:
between LpHq and φpLpHq q. This means in particular that the evaluation map‹‹
pings pLpHq:q Ñ , α ÞÑ αpAq are continuous. This dual pairing between the
spaces pLpHq:, ~¨~L q and pS1 pHq: , ~¨~1 q provides us also with the required dual
pair of cones necessary to construct the mapping LpHq` Ñ L`
8.

R

1Meaning

that φ yields a homeomorphism, i.e., a bijective function with φ and φ´1 continuous,
where φpLpHq: q inherits the subspace topoplogy from pLpHq: q‹‹ .

A NOTE ON HERGLOTZ’S THEOREM FOR TIME SERIES ON FUNCTION SPACES

7

Proposition 3.4. The space LpHq` forms a generating pointed convex cone of
LpHq. Its dual cone is the class of non-negative trace class operators S1 pHq` .
The proof is given in section 5.1. One important technicality is that the topological
embedding (3.3) is in itself not enough to consistently and uniquely identify a LpHq` valued measure with a family of `
8 -valued measures. For this, we require that
`
φpLpHq q is in fact the set of all addition-preserving maps from the dual cone into
`
, i.e., all monoid homomorphisms S1 pHq` Ñ ` where the associative binary
operation is addition. That is, we need that

R

R

R

φpLpHq` q “ HompS1 pHq` ,

R`q

(3.4)

Unlike in finite dimensions, where the topological and algebraic dual coincide and all
semigroup homomorphisms S1 pHq` Ñ ` are automatically continuous, this is not
necessarily the case in infinite dimension. Yet, it is crucial that (3.4) holds. To see
B
this, suppose we have a family of measures tµA uAPS1 pHq` such that A ÞÑ µA P p ` q
is a monoid homomorphism. If (3.4) does not hold, then we have no guarantee
that this family identifies a LpHq` -valued measure. In order for (3.4) to hold, it is
however sufficient that the monoid homomorphisms S1 pHq` Ñ ` are continuous.
It can be shown that positive functionals on a Banach space are in fact continuous
(Neeb, 1998, Prop I.7). Since for every B P LpHq` , the functional φpBq is a positive
functional on S1 pHq: it therefore must be continuous. Suppose we now have an
element of α P HompS1 pHq` , ` q. Continuity of α together with the fact that
S1 pHq` is generating the space S1 pHq: , implies by the Hahn-Banach theorem that
it extends uniquely to a linear functional S1 pHq: Ñ . But this means it can be
represented by an element B P LpHq` , i.e., φpBq “ α. Therefore we have that (3.4)
is satisfied. Furthermore, φpLpHq` q can be shown to have a dense image (Glockner,
2003) in the set
Hommon pS1 pHq` , `
8 q,
2
which is a compact topological monoid of monoid homomorphisms from S1 pHq`
into the compact additive monoid `
8 . That this set is a compact topological monoid
follows by noting that Hommon pS1 pHq` , `
8 q can be seen as a closed vector subspace
`
` S1 pHq
and therefore inherits the topology of pointwise conof the compact set p 8 q
3
`
vergence on S1 pHq . Addition is therefore continuous w.r.t. to this topology.
Moreover being a closed subset of a compact set means it is compact. It is therefore
natural to define the compactification of LpHq` by this this set, i.e.,

R

R

R

R

R

R

R

R

R

`
L`
8 :“ Hommon pS1 pHq ,

R`8q.

(3.5)

3.1.2. Construction of LpHq` -valued measures
With the compactification given by (3.5), we formally define a LpHq` -valued measure as follows.
Definition 3.5. Let pV, Bq be a measurable space. A LpHq` -valued measure is a
countably additive function
µ : B Ñ L`
8
with µpHq “ OH . The measure µ is called a finite LpHq` -valued measure if
µpV q P Hommon pS1 pHq` , ` q – LpHq` . It is σ-finite if E is the countable union of

R

2It

is a monoid as a set and the monoid operations are continuous.
that this corresponds to the weak operator topology on LpHq: .

3Note

8

ANNE VAN DELFT AND MICHAEL EICHLER

Ť8

measurable sets with finite measure, i.e., if E “
for all i P .

N

i“1

Ei and where µpEi q P LpHq`

Using (3.3), we construct the measures uniquely based on a family of positive
measures that depend additively on elements A P S1 pHq` . Fix an element E P B,
and let µpEq P L`
8 . By construction, we can identify with µpEq all maps
A ÞÑ µpEqpAq :“ µA pEq “ TrpµpEqAq,

A P S1 pHq` .

`
It is clear that µA : B Ñ R8
is a countable additive positive measure for all A P
`
S1 pHq. That is, A ÞÑ µpEqpAq is a homomorphism between monoids with respect
to addition, i.e., depends additively on A; For every A1 , A2 P S1 pHq`

µpEqpA1 ` A2 q “ TrpµpEqpA1 ` A2 qq “ TrpµpEqA1 q ` TrpµpEqA2 q
“ µpEqpA1 q ` µpEqpA2 q;
µpEqpOH q “ TrpµpEqOH q “ 0.
On the other hand, if we have a family of positive measures pνA qAPS p Hq` on pV, Bq
1
B
such that, for fixed E P B, A ÞÑ νA pEq P p `
8 q is a monoid homomorphism. It
`
is clear from the definition of B8
that there exists a unique µpEq P L`
8 such that
µpEqpAq “ νA pEq for all A P S1 pHq` . Countable additivity is direct from noting
that elements of the monoid (3.5) can be seen as continuous functionals on S1 pHq`
w.r.t. the topology inherited from S1 pHq` , we therefore have

R

µ

8
`ď
i

˘

Ei pAq “ νA p

8
ď

Ei q “ lim

i

N Ñ8

N
ÿ

νA pEi q “

i“1

8
ÿ

µpEi qpAq

i“1

for all A P S1 pHq` . The following theorem summarizes this and is a simplification
of Theorem I.10 of Neeb (1998).
Theorem 3.6. Let tµA uAPS1 pHq` be a family of non-negative measures on the measurable space pV, Bq s.t. for each borel set E Ď V the assignment A ÞÑ µA pEq
is a monoid homomorphism. Then there exists for each E P B a unique element
`
`
µpEq P L`
8 with µpEqpAq “ µA pEq for all A P S1 pHq and the function µ : B Ñ L8
`
is a LpHq -valued measure.
In particular, it follows that a L`
8 -valued Radon measure µ on a locally compact
space V is a measure with the property that, for each non-negative trace class
operator A P S1 pHq` , the measure µA : B Ñ `
8 is a finite non-negative Radon
measure on V . Moreover, integrability of a function with respect to the scalar
measure furthermore implies integrability of the function with respect to the LpHq` valued measure.

R

3.2. Functional Herglotz’s theorem
We are now ready to prove the following generalization of Theorem 3.1.
Theorem 3.7 (Functional Herglotz’s Theorem). A function Γ :
non-negative definite if and only if
żπ
Γphq “
eihω dF pωq
hP ,
´π

Z

Z Ñ LpHq is
(3.6)

where F is a right-continuous, non-decreasing, finite L`
8 -valued measure on r´π, πs
with F p´πq “ OH . The finite measure F is uniquely determined by Γphq, h P .

Z

A NOTE ON HERGLOTZ’S THEOREM FOR TIME SERIES ON FUNCTION SPACES

9

Proof. Suppose first that Γphq admits the representation (3.6) with respect to a
right-continuous, non-decreasing, finite L`
8 -valued measure F on r´π, πs. Note
first that since F is a finite measure, the integral is well-defined. Let us denote
the representation of this integral by ν : Z Ñ LpHq. Since Γphq: “ Γp´hq, it is
clear that Γp¨q is hermitian. By (ii) of Definition 3.2, it is sufficient to show that the
operator-valued kernel cph1 , h2 q :“ νph1 h‹2 q “ νph1 ´ h2 q is non-negative definite.
Using (i) of Definition 3.2,
n
ÿ
xνphj ´ hk qgk , gj y
j,k“1

ż
n
ÿ
“
x
j,k“1

Z

“

ż

“

ż

eiphj ´hk qω dµpωqgk , gj y
r´π,πs
n
ÿ

xeiphj ´hk qω gk , gj ydµpωq

r´π,πs j,k“1

r´π,πs

n
›2
›ÿ
›
eihk ω gk › dµpωq ě 0
2

k“1

for all h1 , . . . , hn P and g1 , . . . , gn P H.
Conversely, suppose that Γp¨q is a LpHq` -valued function on and let A be a
an element of S1 pHq` . Define the function ΓA : Ñ
by ΓA phq :“ TrpΓphqAq.
Since the square root of both Γphq and A are well-defined and the trace satisfies
TrpΓphqAq “ TrpAΓphqq, it is direct that ΓA is a non negative definite function on
the integers. By Theorem 3.1, we therefore have the representation
żπ
CA phq “
eihω dFA pωq
hP ,

Z

´π

C

Z

Z

where FA p¨q with FA p´πq “ 0 is a uniquely determined Radon measure on r´π, πs.
The measure is finite since CA p q “ CA p0q “ FA pr´π, πsq ă 8. A similar derivation
as in the previous subsection demonstrates that A ÞÑ FA pEq is a monoid homomorphism S1 pHq` Ñ ` with respect to addition for each E P B. By Theorem 3.6,
the family of measures tFA uAPS1 pHq` therefore uniquely identifies a finite measure
F pEq P L`
8 on the measurable space pr´π, πs, Bq with F pEqpAq “ TrpF pEqAq “
FA pEq for all A P S1 pHq` . This completes the proof.


Z

R

er representation
4. A generalized fun tional Cram
The Spectral Representation Theorem (Cramér, 1942), often called the Cramér representation, is as fundamental to frequency domain analysis as Wold’s representation is to the time domain. It asserts that every (finite-dimensional) zero-mean
weakly stationary process can be represented as a superposition of sinusoids with
random amplitudes and phases that are uncorrelated. An important ingredient in
establishing this classical theorem is the existence of an isometric isomorphism that
allows to identify a weakly stationary time series on the integers with an orthogonal
increment process on r´π, πs. As already mentioned, an initial generalization of
the Cramér representation to functional-valued weakly stationary time series was
first considered by Panaretos
and Tavakoli (2013a), but is restricted to processes for
ř
which the assumption hPZ ~Ch ~1 ă 8 holds. In this section, we shall use the established functional Herglotz’s theorem (Theorem 3) to derive a functional Cramér

10

ANNE VAN DELFT AND MICHAEL EICHLER

representation that can be seen as a true generalization of the classical theorem to
the function space. For this, we shall make the following assumption on the secondorder dependence structure, which we believe is not just a natural assumption for
L2 -valued processes4 but is in fact a necessary assumption for a frequency domain
representation to exist.

Z

Assumption 4.1. tXt : t P u is a weakly stationary H-valued time series with lag
covariance operators tCh uhPZ P S1 pHq.
In order to establish a spectral representation under this assumption, we will show
that the mapping
Xt ÞÑ eit¨

P

forms an Hilbert space isometric isomorphism between L2H pΩ, q and L2 pr´π, πs, µF q
where we define
µF pEq :“ ~F pEq~1
(4.1)
for all Borel sets E Ď r´π, πs. Here, F is the operator-valued measure on r´π, πs
induced by the sequence of covariance operators tCh uhPZ of tXt : t P u. Before we
derive the properties of the mapping, we have to verify that this indeed defines a
measure. This is the contents of the following lemma.

Z

Lemma 4.2. Let Assumption 4.1 be satisfied. Then the function µF defined in (4.1)
is a finite scalar-valued measure on r´π, πs.

Z

Proof of Lemma 4.2. By Proposition 3.3, the covariance function Cp¨q : Ñ S1 pHq of
a weakly stationary H-valued time series is non-negative definite. Under assumption
4.1, Theorem 3 implies this function uniquely determines a S1 pHq` -valued measure
F on r´π, πs . Using the properties of F , it is now straightforward to verify
that the function µF : B Ñ r0, 8s is a non-negative scalar-valued measure on
the measurable space pr´π, πs, Bq. Firstly, for each Borel set E Ď r´π, πs and
every e P H, we have xF pEqe, ey ě 0 and thus µF “ TrpF q ě 0. Secondly,
xF pHqe, ey “ xOH e, ey “ µF pHq “ 0 which follows by definition of F . Thirdly, for
all countable selections of pairwise disjoint set tEi uiPN in B, countable additivity of
F yields
µF

8
`ď

˘

Ei “

i“1

8
ÿ
j

xF p

8
ď

i“1

8 ÿ
8
ÿ
Ei qej , ej y “
x F pEi qej , ej y,
j“1

i

where tej ujPN is an orthonormal basis of H. By continuity of the inner product and
the fact that F pr´π, πsq ă 8, Fubini’s theorem implies
8 ÿ
8
ÿ

i“1 j“1

xF pEi qej , ej y “

8
ÿ

i“1

~µF pEi q~1 “

8
ÿ

µF pEi q

i“1

and thus µF is countably additive. Finally, since F is S1 pHq` -valued measure on
r´π, πs it is direct that µF pr´π, πsq ă 8.

Additionally, to be able to properly define the spectral representation we require
the notion of a H-valued orthogonal increment process.
4As

explained in section 2.2, it is reasonable to assume that the lag covariance operators belong
to the same Schatten-class as C0 , which is the class S1 pHq by Parseval’s Identity.

A NOTE ON HERGLOTZ’S THEOREM FOR TIME SERIES ON FUNCTION SPACES

11

Definition 4.3. A H-valued random process tZω : ´ π ď ω ď πu defined on a
probability space pΩ, A, q is a functional orthogonal increment process, if for all
g1 , g2 P H and ´π ď ω ď π
(i) the operator pZω b Zω q is an element of S1 pHq`
(ii) xZω , g1y “ 0
`
˘
pω1 , ω2 s X pω3 , ω4 s “ H.
(iii) x pZω4 ´ Zω3 q b pZω2 ´ Zω1 q g1 , g2y “ 0,
`
˘
(iv) x pZω`ε ´ Zω q b Zω`ε ´ Zω g1 , g2 y Ñ 0, as ε Ó 0.

P

E
E
E

E

H

To establish the isomorphism, let H Ă
denote the space spanned by all finite
linear combinations of the random functions Xt , i.e., H “ sptXt : t P u. We remark
that the inner product on satisfies

H

Z

E

xX1 , X2 yH “ Trp pX1 b X2 qq,
Furthermore let the space H denote the space of all square-integrable functions
on r´π, πs with respect to the measure µF , i.e., H “ L2 pr´π, πs, µF q. This space
becomes a Hilbert space once we endow it with the inner product
żπ
żπ
Ę F pωq “ Trp
xf, gyH “
f pωqgpωqdµ
f pωqgpωqdF pωqq
f, g P H,
´π

´π

where the last equality follows by non-negative definiteness of F and linearity of
the trace operator.
Theorem 4.4. Let F be the S1 pHq` -valued measure corresponding to the process tXt u. Then there exists an isometric isomorphism T between sptXt u and
L2 pr´π, πs, µF q such that
T Xt “ eit¨ ,
tP .
The process defined by
`
˘
Zω “ T ´1 1p´π,ωs p¨q
is then a functional orthogonal increment process of which the covariance structure
is uniquely determined by F and satisfies
“
‰
E pZω ´ Zλ q b pZω ´ Zλ q “ F pωq ´ F pλq, ´π ď λ ď ω ď π

Z

Proof of Theorem 4.4. Consider first the mapping T : H Ñ H given by
n
n
ÿ
ÿ
T p aj Xtj q “
aj eit¨
tP

Z

j“1

j“1

It is straightforward
to seeř
the mapping is linear and preserves inner products. Let
ř
Y “ nj“1 aj Xtj and W “ nj“1 bj Xtj . By Theorem 3,
xT Y, T W yH “
“
“
“

n
ÿ

i,j“1
n
ÿ

i,j“1
n
ÿ

i,j“1
n
ř

i,j“1

ai bj xei¨ti , ei¨tj yH
ai bj

żπ

eiλpti ´tj q dµF pλq

´π

ai bj Trp

żπ

eiλpti ´tj q dF pλqq

´π

ai bj TrpCi´j q “

n
ř

i,j“1

ai bj xXti , Xtj yH “ xY, W yH

12

ANNE VAN DELFT AND MICHAEL EICHLER

For the extension of the isomorphism over the closure of H onto the closure of H ,
note that if Y is an element of H̄ then there must exist a sequence tYn uně1 P H
converging to Y . Denote T pY q to be the limit of T pYn q, i.e.,
T pY q :“ lim T pYn q
nÑ8

Since tYn u is a Cauchy sequence and T norm-preserving, tT Yn u is a Cauchy sequence
1
in L2 pr´π, πs, µF q and thus T pY q P H¯ . If there is another sequence tYn u P H
converging to Y , then the limit must be unique since
1

1

lim }T pYn q ´ T pYn q}H “ lim }Yn ´ Yn }H “ 0,

nÑ8

nÑ8

and therefore the extension is well-defined. Preservation of linearity and the isometry property are straightforward from linearity of T on H and continuity of the inner
product, respectively. To show that the closure of H is in fact L2 pr´π, πs, µF q, we
recall that the Stone-Weierstrass theorem (Fejer’s theorem) implies that H is dense
in the space of 2π-periodic continuous functions on r´π, πs. Moreover, by Proposition 4.2, µF is a finite Radon measure (i.e., finite and regular) on r´π, πs. The set
of continuous functions with compact support are therefore in turn uniformly dense
in L2 pr´π, πs, µF q (see e.g., Bogachev, 2006; Rudin, 1987). Consequently we find
H¯ “ L2 pr´π, πs, µF q. The inverse mapping T ´1 : L2 pr´π, πs, µF q Ñ H̄ is therefore
properly defined. This finishes the proof of the first part of the theorem.
Les us then define, for any ω P p´π, πs, the process
`
˘
Zω “ T ´1 1p´π,ωs p¨q

with Z´π ” 0 P H. By the established isometry, this process is well-defined in H̄.
Therefore there must exist a sequence tYn u in H such that limnÑ8 }Yn ´ Zω } “ 0.
Since all elements in the sequence have zero-mean, continuity of the inner product
implies

E

E

xZω , f yH “ lim Trp pYn b f qq “ lim x rYns, f y “ 0 @f ı 0 P H
nÑ8

nÑ8

showing the process tZω : ´π ď ω ď πu has zero mean. Additionally,
xZω4 ´ Zω3 , Zω2 ´ Zω1 yH “ x1pω3 ,ω4 s p¨q, 1pω1 ,ω2 s p¨qyH
żπ
“
1pω3 ,ω4 s pωq1pω1 ,ω2 s pωqdµF pωq

(4.2)

´π

For all pω1 , ω2 s X pω3 , ω4 s “ H, this inner product is zero while for ω3 “ ω1 , ω4 “ ω2
we have
ω1 ď ω2
xZω2 ´ Zω1 , Zω2 ´ Zω1 yH “ µF pω2 q ´ µF pω1 q.
showing that the tZω u is right-continuous. We can also write (4.2) as
ż
` π
˘
Tr
1pω3 ,ω4 s pωq1pω1 ,ω2 s pωqdF pωq .
´π

For ω3 “ ω1 , ω4 “ ω2 this implies

EpZω

2

´ Zω1 q b pZω2 ´ Zω1 q “ F pω2 q ´ F pω1 q.

ω1 ď ω2

where the equality holds in ~¨~1 . The second order structure of Zω is therefore
uniquely defined by the operator valued measure F of the process X.

The generalization of the Cramér representation to processes with possible discontinuities in the spectral distribution is then as follows

A NOTE ON HERGLOTZ’S THEOREM FOR TIME SERIES ON FUNCTION SPACES

13

Theorem 4.5 (Functional Cramér representation). Suppose tXt u satisfies Assumption 4.1. Then there exists a right-continuous functional orthogonal increment
process tZω , ´π ď ω ď πu with Z´π ” 0 P H such that
żπ
Xt “
eitω dZω
a.s.
´π

Proof of Theorem 4.5. Consider the subspace Hs of H¯ containing the simple functions, i.e., the space Hs contains elements of the form
n
ÿ
gpωq “
ai 1p´ωi ,ωi`1 s
i“1

for a partition Pn “ t´π “ ω0 ă ω1 ă ¨ ¨ ¨ ă ωn`1 “ πu of r´π, πs and ai P
define the mapping I : Hs Ñ H̄ given by
n
ÿ
Ipgq “
ai pZωi`1 ´ Zωi q

C. Then

i“0

By Theorem 4.4, this is an isomorphism from H¯ onto H̄ and coincides with T ´1 .
More specifically, Ipei¨t q “ T ´1 pei¨t q “ T ´1 T pXt q “ Xt and the statement of the
Theorem follows by taking the Riemann-Stieltjes integral limit
n
ÿ
›
›
›Xt ´
eiωi t pZωi`1 ´ Zωi q›H Ñ 0, as n Ñ 8,
i“0

where meshpPn q Ñ 0 as n Ñ 8. More generally, for any g P L2 pr´π, πs, µF q,
the mapping Ipgq corresponds to the Riemann-Stieltjes integral with respect to the
othogonal increment process Zω .

Acknowledgements. This work has been supported in part bu the Collaborative
Research Center “Statistical modeling of nonlinear dynamic processes” (SFB 823,
Project A1, C1, A7) of the German Research Foundation (DFG).
5. Auxiliary derivations
5.1. Dual pair of cones
Proof of Proposition 3.4. We start by remarking that positive elements on a C ‹
algebra form a closed convex cone ta‹ a : a P C ‹ u(see e.g. Conway, 1990). Since
elements of LpHq` have a positive square root, LpHq` forms a pointed convex cone
in the Banach algebra of bounded linear operators. The topological dual cone to a
cone C of a real vector space V can be defined as the set
1

tv P V : xv, cy ě 0 @c P Cu
1

where x¨, ¨y denotes the duality pairing between V and its topological dual V . With
a similar argument as abve, it is straightforward to verify that CV :“ S1 pHq` is a
pointed convex cone in the Banach space V :“ pS1 pHq:, ~¨~1 q. From section 3.1.1
1
and the above, its topological dual is the Banach space V :“ pLpHq: , ~¨~L q with
1
cone CV 1 :“ pLpHq` , ~¨~L q in V . We also know that we can identify an element of
a Banach space with an element from its bidual. More specifically, we have that V
1
1
1
1
canonically embeds into pV q i.e., V Ď pV q Ñ CV Ď pCV 1 q1 . Since the cone CV 1
is closed, the Hahn-Banach separation theorem implies that CV “ pCV 1 q1 , which

14

ANNE VAN DELFT AND MICHAEL EICHLER

identifies the cone of non-negative trace class operators as the dual cone of the
non-negative bounded linear operators.


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