1. Introduction
Let
,
, be a
d-dimensional weakly stationary time series, where each
is a complex-valued random variable on the same probability space
. It is a second-order, and in this sense, translation-invariant process:
where
,
, is the non-negative definite covariance matrix function of the process. (Here and below, if
is a matrix then
denotes its adjoint matrix, i.e., its complex conjugate transpose. Vectors like
are written as column matrices, so
is a row matrix. All the results are valid for real-valued time series as well with no change; then
denotes matrix transpose.) Without loss of generality, from now on it is assumed that
.
Thus, the considered random variables will be d-dimensional, square integrable, zero expectation random complex vectors whose components belong to the Hilbert space . The orthogonality of the random vectors and is defined by the relationship .
The
past of until time is the closed linear span in
of the past and present values of the components of the process:
The remote past of is . The process is called regular if and it is called singular if . Of course, there is a range of cases between these two extremes.
Singular processes are also called
deterministic (see, e.g.,
Brockwell et al. 1991) because based on the past
, future values
,
, …, can be predicted with zero mean square error. On the other hand, regular processes are also called
purely non-deterministic, since their behavior is completely influenced by random
innovations. Consequently, knowing
, future values
,
, …, can be predicted only with positive mean square errors
,
, …, and
. This shows why studying
regular time series is a primary target both in the theory and applications. The Wold decomposition proves that any non-singular process can be decomposed into an orthogonal sum of a nonzero regular and singular process. This also supports why it is important to study regular processes.
The Wold decomposition implies (see, e.g., the classical references
Rozanov 1967 and
Wiener and Masani 1957) that
is regular if and only if
can be written as a causal infinite moving average (
a Wold representation)
where
is an
r-dimensional
orthonormal white noise sequence
,
,
is the
identity matrix. If the white noise process
in (
1) is given by the Wold representation, then it is unique up to a multiplication by a constant unitary matrix; therefore, it is called
a fundamental process of the regular time series. In this case the pasts of
and
are identical:
for any
.
An important use of Wold representation is that
the best linear h-step ahead prediction can be given in terms of that. If the present time is 0, then the orthogonal projection of a future random vector
(
) to the Hilbert subspace
representing past and present is
Then gives the best linear prediction of with minimal least square error.
An alternative way to write Wold representation is
where
is the
d-dimensional white noise process of
innovations:
where
denotes the orthogonal projection of the random vector
to the Hilbert subspace
. Furthermore,
,
,
is a
non-negative definite matrix of rank
r,
, the covariance matrix of the best linear one-step ahead prediction error.
It is also classical that any weakly stationary process has a non-negative definite spectral measure matrix
on
such that
Then
is regular (see again, e.g.,
Rozanov 1967 and
Wiener and Masani 1957) if and only if
, the spectral density
has a.e. constant rank
r, and can be factored in a form
where
denotes spectral norm. Here the sequence of coefficients
is not necessarily the same as in the Wold decomposition. Furthermore,
so the entries of the
analytic matrix function are analytic functions in the open unit disc
D and belong to the class
on the unit circle
T; consequently, they belong to the Hardy space
. It is written as
or briefly
.
Recall that the Hardy space
,
, denotes the space of all functions
g analytic in
D whose
norms over all circles
,
, are bounded; see, e.g., (
Rudin 2006, Definition 17.7). If
, then equivalently,
is the Banach space of all functions
such that
so the Fourier series of
is one-sided,
when
; see, e.g., (
Fuhrmann 2014, sct. II.12). Notice that in formulas (
6) and (
7) there is a negative sign in the exponent; this is a matter of convention that I am going to use in the sequel too.
An especially important special case of Hardy spaces is
, which is a Hilbert space, and which by Fourier transform is isometrically isomorphic with the
space of sequences
with norm square
For a one-dimensional time series
(
) there exists a rather simple sufficient and necessary condition of regularity given by (
Kolmogorov 1941):
- (1)
has an absolutely continuous spectral measure with density f;
- (2)
, that is, .
Then the Kolmogorov–Szeg
formula also holds:
where
is the variance of the innovations
, that is, the variance of the one-step ahead prediction.
For a multidimensional time series
which has
full rank, that is, when
has a.e.
full rank:
, and so the innovations
defined by (
4) have full rank
d, there exists a similar simple sufficient and necessary condition of regularity; see
Rozanov (
1967) and
Wiener and Masani (
1957):
- (1)
has an absolutely continuous spectral measure matrix with density matrix ;
- (2)
, that is, .
Then the
d-dimensional Kolmogorov–Szeg
formula also holds:
where
is the covariance matrix of the innovations
defined by (
4).
Unfortunately, the generic case of regular time series when the rank of the process can be smaller than the dimension is rather complicated. To the best of my knowledge, in that case, (
Rozanov 1967, Theorem 8.1) gives a necessary and sufficient condition of regularity. Namely, a
d-dimensional stationary time series
is regular and of rank
r,
, if and only if each of the following conditions holds:
- (1)
It has an absolutely continuous spectral measure matrix with density matrix which has rank r for a.e. .
- (2)
The density matrix
has a principal minor
, which is nonzero a.e. and
- (3)
Let denote the determinant obtained from by replacing its ℓth row by the row . Then the functions are boundary values of functions of the Nevanlinna class N.
It is immediately remarked in the cited reference that “unfortunately, there is no general method determining, from the boundary value of a function , whether it belongs to the class N”.
Recall that the Nevanlinna class
N consists of all functions
g analytic in the open unit ball
D that can be written as a ratio
,
,
,
, where
and
denote Hardy spaces; see, e.g., (
Nikolski 2019, Definition 3.3.1).
The aim of this paper is to extend from the one-dimensional case to the multidimensional case a well-known sufficient condition for the regularity and a method of finding an spectral factor and the covariance matrix in the case of smooth spectral density.
2. Generic Regular Processes
To find an spectral factor if possible, a simple idea is to use a spectral decomposition of the spectral density matrix. (Take care that here we use the word ‘spectral’ in two different meanings. On one hand, we use the spectral density of a time series in terms of a Fourier spectrum, and on the other hand we take the spectral decomposition of a matrix in terms of eigenvalues and eigenvectors.)
So let
be a
d-dimensional stationary time series and assume that its spectral measure matrix
is absolutely continuous with density matrix
which has rank
r,
, for a.e.
. Take the parsimonious spectral decomposition of the self-adjoint, non-negative definite matrix
:
where
for a.e.
, is the diagonal matrix of nonzero eigenvalues of
and
is a sub-unitary matrix of corresponding right eigenvectors, not unique even if all eigenvalues are distinct. Then, still, we have
The matrix function
is a self-adjoint, positive definite function, and
where
is the density function of a finite spectral measure. This shows that the integral of
over
is finite. Thus
can be considered the spectral density function of a full rank regular process. So it can be factored, and in fact, we may take a miniphase
spectral factor
of it:
where
is a diagonal matrix.
Then a simple way to factorize
is to choose
where
, each
being a measurable function on
such that
for any
, but otherwise arbitrary and
still denotes a sub-unitary matrix of eigenvectors of
in the same order as the one of the eigenvalues.
To the best of my knowledge, it is not known if for any regular time series
there exists such a matrix-valued function
such that
defined by (
13) has a Fourier series with only non-negative powers of
. Equivalently, does there exist an
analytic matrix function
whose boundary value is the above spectral factor
with some
, according to Formulas (
5)–(
7)?
Thus, the role of each
is to modify the corresponding eigenvector
so that
has a Fourier series with only non-negative powers of
, this way achieving that
. Equivalently, if
is the boundary value of a complex function
defined in the unit disc,
, and
has singularities for
, then we would like to find a complex function
in the unit disc so that
and
is analytic in the open unit disc
D and continuous in the closed unit disc
,
Carrying out this procedure for , one would obtain an sub-unitary matrix function.
Example 2.2.4 in (
Szabados 2022) shows that—at least theoretically—this task can be carried out in certain cases. Furthermore, as a very similar problem, in the case of a rational spectral density, one can always find an inner matrix multiplier so that it gives a rational analytic matrix function
whose boundary value is an
spectral factor
; see, e.g., (
Rozanov 1967, chp. I, Theorem 10.1).
Theorem 1. - (a)
Assume that a d-dimensional stationary time series is regular of rank r, . Then for defined by (
10)
one has , equivalently, - (b)
If, moreover, one assumes that the regular time series is such that it has an spectral factor of the form (
13)
, then the following statement holds as well: The sub-unitary matrix function appearing in the spectral decomposition of in (
9)
can be chosen so that it belongs to the Hardy space , and thus In this case one may call an inner matrix function.
The next theorem gives a sufficient condition for the regularity of a generic weakly stationary time series; compare with the statements of Theorem 1 above. Observe that assumptions (1) and (2) in the next theorem are necessary conditions of regularity as well. Only assumption (3) is not known to be necessary. We think that these assumptions are simpler to check in practice than those of Rozanov’s theorem cited above. By formula (
13), checking assumption (3) means that for each eigenvector
of
we are searching for a complex function multiplier
of unit absolute value that gives an
function result.
Theorem 2. Let be a d-dimensional time series. It is regular of rank if the following three conditions hold.
- (1)
It has an absolutely continuous spectral measure matrix with density matrix which has rank r for a.e. .
- (2)
For defined by (
10)
one has ; equivalently, (
15)
holds. - (3)
The sub-unitary matrix function appearing in the spectral decomposition of in (
9)
can be chosen so that it belongs to the Hardy space ; thus, (
16)
holds.
Next we discuss a multivariable version of a well-known one-dimensional theorem; see, e.g., (
Lamperti 1977, sct. 4.4). This theorem gives the Wold representation of a regular time series
. First let us recall some facts we are going to use. A sufficient and necessary condition of regularity is given by the factorization (
5) and (
6) of the spectral density
, where the
spectral factor
is in
. Using Singular Value Decomposition (SVD), we can write that
where
is a
sub-unitary matrix,
is an
unitary matrix,
is an
diagonal matrix of positive singular values
, for a.e.
. Clearly,
, for
.
The (generalized) inverse of
is not unique when
. Let
be the Moore–Penrose inverse of
:
We also need the spectral (Cramér) representation of the stationary time series
where
is a stochastic process with orthogonal increments, obtained by the
isometry between the Hilbert spaces
and
; see, e.g., (
Bolla and Szabados 2021, sct. 1.3). Namely, if
(
), then
Theorem 3. Assume that the spectral measure of a d-dimensional weakly stationary time series is absolutely continuous with density which has constant rank r, . Moreover, assume that there is a finite constant M such that for all , and has a factorization , where and its Moore–Penrose inverse as well.
Then the time series is regular and its fundamental white noise process can be obtained aswhereis the Fourier series of , convergent in sense. The sequence of coefficients of the Wold representation is given by the convergent Fourier series Proof. The regularity of
obviously follows from the assumptions by (
5) and (
6).
First let us verify that the stochastic integral (
19) is correct, that is, the components of
belong to
:
This also justifies that
Second, let us check that the sequence defined by (
19) is orthonormal, using the isometry (
18):
Third, let us show that
is orthogonal to the past
for any
:
for any
, since
, so its Fourier coefficients with negative indices are zero.
Fourth, let us see that for . Because of stationarity, it is enough to show that . Since is the closure in of the components of all finite linear combinations of the form , by the isometry it is equivalent to the fact that belongs to the closure in of all finite linear combinations of the form .
Using the assumed boundedness of
, we obtain that
We assumed that
, which means that
has a one-sided Fourier series (
21) which converges in
, where
denotes Lebesgue measure:
Since the spectral norm squared of a matrix is bounded by the sum of the absolute values squared of the entries of the matrix, these imply that the last term in (
23) tends to 0 as
. This shows that
.
Fifth, by (
17), we see that
a.e. in
. Consequently, the difference
is orthogonal to itself in
, so it is a zero vector. Then by (
19) and (
22),
Equation (
24) shows that each entry of
belongs to
, so the isometry between this space and
justifies (
25).
Finally, the previous steps show that the innovation spaces of the sequences
and
are the same for any time
, so the pasts
and
agree as well for any
. Thus (
25) gives the Wold representation of
. □
3. Smooth Eigenvalues of the Spectral Density
In the one-dimensional case there is a well-known sufficient condition of regularity, which at the same time gives a formula for an
spectral factor and also for the white noise sequence and the coefficients in the Wold decomposition (
1). This is the assumption that the process has a continuously differentiable spectral density
for any
; see, e.g., (
Lamperti 1977, p. 76) or (
Bolla and Szabados 2021, sct. 2.8.2).
This sufficient condition can be partially generalized to the multidimensional case. When a regular
d-dimensional time series
has an
spectral factor of the form (
13), equivalently, it has a sub-unitary matrix function
appearing in the spectral decomposition of
in (
9) that can be chosen so that it belongs to the Hardy space
, and therefore the smoothness of the nonzero eigenvalues of the spectral density
gives a formula for an
spectral factor. The argument above in the paragraph around Equation (
14) shows that in certain cases one can find such a sub-unitary matrix function
.
Theorem 4. Let be a d-dimensional time series. It is regular of rank if the following three conditions hold.
- (1)
It has an absolutely continuous spectral measure matrix with density matrix which has rank r for a.e. .
- (2)
Each nonzero eigenvalue of is a continuously differentiable positive function on .
- (3)
The sub-unitary matrix function appearing in the spectral decomposition of in (
9)
can be chosen so that it belongs to the Hardy space , and thus (
16)
holds.
Moreover, satisfies the conditions of Theorem 3 too, so formulas (
19)–(
22)
give the Wold representation of . Proof. Condition (2) implies that each
is also a continuously differentiable function on
and so it can be expanded into a uniformly convergent Fourier series
Observe that each and consequently each is a continuous function on , so it is in . Moreover, each and consequently each has only positive powers of in its Fourier series. So each belongs to the Hardy space .
Substitute (
28) into (
9):
Thus we can take a spectral factor
Since each
and by condition (3) each entry of
is in
, each entry of
is in
. It means that
is an
spectral factor as in (
5) and (
6), and consequently
is regular.
Take the Moore–Penrose inverse of
:
where by (
27) each
so it is also continuous and its Fourier series has only positive powers of
too. It implies that
.
Finally, since each is a continuous function on , and thus bounded, and the components of are bounded functions because is sub-unitary, it follows that is bounded. □
The
d-dimensional Kolmogorov–Szeg
Formula (
8) gives only the determinant of the covariance matrix
of the innovations in the full rank regular time series. Similar is the case when the rank
r of the process is less than
d:
where
is the diagonal matrix of the
r nonzero eigenvalues of
and
is the covariance matrix of the innovation of an
r-dimensional subprocess of rank
r of the original time series; see (
Bolla and Szabados 2021, Corollary 4.5) or (
Szabados 2022, Corollary 2.5)
Fortunately, under the conditions of Theorem 4, one can obtain the covariance matrix
itself by a similar formula, as the next theorem shows.
Theorem 5. Assume that a weakly stationary d-dimensional time series satisfies the conditions of Theorem 4. Then the covariance matrix Σ of the innovations of the process can be obtained aswhere , , are the nonzero eigenvalues of the spectral density matrix of the process, is the matrix of corresponding orthonormal eigenvectors, and Proof. The error of the best 1-step linear prediction by (
2), and the same time, the innovation is
using the Wold decomposition of
. Thus the covariance of the innovation is
With the analytic function
corresponding to the Wold decomposition by (
7),
. Taking the Fourier series (
16), let
In addition, denote by (
27)
Now using (
29), it follows that
and
Combining the previous results,
where
is given by (
30) and by (
26),
This completes the proof of the theorem. □