MATHEMATICS

Yu. A. ROZANOV

ON STATIONARY SEQUENCES FORMING A BASIS

(Presented by Academician A. N. Kolmogorov on 27 X 1959)

In linear prediction of a multidimensional stationary random process
(x(t)={x_1(t),\ldots,x_n(t)}) (the time (t) takes only integer values), the question of the possibility of representing the quantities of the best prediction in the form of a series

[
h=\sum_{k=1}^{n}\sum_{t\in T} c_k(t)x_k(t)
\tag{1}
]

(convergent in the mean square) in terms of the values (x_k(t)) observed at time instants (t\in T) is very important.

Let (H) denote the linear closure in the mean square of the quantities (x_k(t)), (k=1,\ldots,n), (-\infty<t<\infty). As usual, we identify all random variables (h) that differ from one another only with probability zero, and introduce in (H) the scalar product ((h_1,h_2)=Mh_1\overline{h_2}).

The question of representability of random variables (h\in H) in the form of the series (1) is the question of when the system of quantities ({x_k(t)}), (k=1,\ldots,n), (-\infty<t<\infty), forms a basis in the Hilbert space (H). In considering this question it is natural to restrict oneself to the case when the system ({x_k(t)}) is minimal, i.e. no quantity (x_k(t)) belongs to the linear closure of the remaining quantities of this system.

Let the process (x(t)) have spectral density (f(\lambda)={f_{kj}(\lambda)}_{k,j=1,\ldots,n}). From work ((^2)) (cf. ((^1))) follows Theorem 1.

Theorem 1. In order that the system ({x_k(t)}) be minimal, it is necessary and sufficient that

[
\int_{-\pi}^{\pi}\frac{1}{\operatorname{Sp} f(\lambda)}\,d\lambda<\infty,
\tag{2}
]

where

[
\operatorname{Sp} f(\lambda)=\sum_{k=1}^{n} f_{kk}(\lambda)
]

is the trace of the spectral density (f(\lambda)).

As is known, the system ({x_k(t)}) is minimal if and only if there exists in the space a conjugate system of quantities ({y_k(t)}), i.e. one such that

[
(x_k(t),y_j(s))=
\begin{cases}
1 & \text{if } k=j,\ t=s;\
0 & \text{if } k\ne j \text{ or } t\ne s.
\end{cases}
\tag{3}
]

If the conjugate system ({y_k(t)}) is complete in (H), then each quantity (h\in H) is uniquely determined by the series

[
h\sim \sum_k\sum_t c_k(t)x_k(t),
\tag{4}
]

where (c_k(t)=(h,y_k(t))), and if the series in (4) converges, then its sum is precisely (h).

Let

[
x_k(t)=\int_{-\pi}^{\pi} e^{i\lambda t}\Phi_k(d\lambda), \qquad k=1,\ldots,n,
\tag{5}
]

be the spectral representation of the process (x(t)). Every quantity (h\in H) can be represented in the spectral form (5)

[
h=\int_{-\pi}^{\pi}\sum_{k=1}^{n}\varphi_k(\lambda)\Phi_k(d\lambda),
\tag{6}
]

where the vector function (\varphi_\lambda={\varphi_1(\lambda),\ldots,\varphi_n(\lambda)}) satisfies the condition

[
\int_{-\pi}^{\pi}(\varphi_\lambda,f_\lambda\varphi_\lambda)\,d\lambda<\infty.
\tag{7}
]

Relation (6) gives an isometric correspondence between the space (H) and the space (L^2) of vector functions (\varphi_\lambda) with scalar product
[
(\varphi,\varphi')=\int_{-\pi}^{\pi}(\varphi_\lambda,f_\lambda\varphi'_\lambda)\,d\lambda.
]

By virtue of the minimality condition (2), the matrix function
[
f_\lambda^{-1}={p_{kj}(\lambda)}{k,j=1,\ldots,n}
]
is integrable, and therefore vector functions of the form (e^{i\lambda t} f\lambda^{-1}\delta_k), where (\delta_k={0,\ldots,1,0,\ldots,0}) is the unit vector, belong to the space (L^2).

Obviously, the quantities of the conjugate system ({y_k(t)}) are represented in the form

[
y_k(t)=\frac{1}{2\pi}\int_{-\pi}^{\pi} e^{i\lambda t}\sum_{j=1}^{n}p_{kj}(\lambda)\Phi_j(d\lambda),
\tag{8}
]

whence the completeness of ({y_k(t)}) in the space (H) follows easily; the coefficients (c_k(t)) in the expansion (4) are the Fourier coefficients of the functions (\varphi_k(\lambda)) occurring in the representation (6):

[
c_k(t)=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{-i\lambda t}\varphi_k(\lambda)\,d\lambda.
\tag{9}
]

Following [the work] ((^3)), we shall call the minimal system ({x_k(t)}) Bessel if, for every (h\in H),
[
\sum_k\sum_t |c_k(t)|^2<\infty,
]
and Hilbert if, for any numbers (c_k(t)),
[
\sum_k\sum_t |c_k(t)|^2<\infty,
]
there exists an (h\in H) with such expansion coefficients that (c_k(t)=(h,y_k(t))).

Theorem 2. In order that the system ({x_k(t)}) be Bessel, it is necessary and sufficient that

[
f(\lambda)\ge mI
\tag{10}
]

for some (m>0) for almost all (\lambda).

Proof. As is known ((^3)), if the system ({x_k(t)}) is Bessel, then there exists a constant (C) such that

[
\sum_k\sum_t |c_k(t)|^2\le C|h|^2,
\tag{11}
]

which leads to the relation

[
\int_{-\pi}^{\pi} |\varphi_\lambda|^2\,d\lambda \leq
C\int_{-\pi}^{\pi}(\varphi_\lambda, f_\lambda\varphi_\lambda)\,d\lambda
\tag{12}
]

for any vector-function (\varphi_\lambda\in L^2), whence the assertion of the theorem follows.

The following theorem is proved analogously:

Theorem 3. In order that the system ({x_k(t)}) be a Hilbert system, it is necessary and sufficient that

[
f(\lambda)\leq MI
\tag{13}
]

for some (M<\infty) for almost all (\lambda).

Let us note that in the case of a Hilbert system, for any numbers (c_k(t)), (\sum_k\sum_t |c_k(t)|^2<\infty), the series (4) converges (4). Thus, if the functions (\varphi_k(\lambda)) in the spectral representation (6) of the random variable (h) are square-integrable, then in the case where the system ({x_k(t)}) is Hilbert, the random variable (h) is expanded in the convergent series (4).

Recall (5) that to the best quantities (\hat{x}_k(t,\tau)) of linear extrapolation of the unknown values (x_k(t)), (k=1,\ldots,n), from the known past of the process—the quantities (x_k(s)), (k=1,\ldots,n), (s\leq\tau)—there correspond in the space (L^2) vector-functions

[
\hat{\varphi}k(t,\tau,\lambda)={\hat{\varphi}(\lambda)},}(\lambda),\ldots,\hat{\varphi}_{kn
]

the components (\hat{\varphi}_{kj}(\lambda)) of which form the matrix (\hat{\varphi}(t,\tau,\lambda)),

[
\varphi(t,\tau,\lambda)
=
e^{i\lambda t}
\left[
a(\lambda)-
\sum_{s=0}^{t-\tau-1} e^{-i\lambda s}a_s
\right]
a^{-1}(\lambda),
\tag{14}
]

where

[
a(\lambda)=\sum_{s=0}^{\infty}e^{-i\lambda s}a_s
]

is the boundary value of a maximal analytic matrix of class (H_2),

[
a(\lambda)a^*(\lambda)=2\pi f(\lambda).
\tag{15}
]

The minimality condition (2) guarantees the square-integrability of the elements of the matrix (a^{-1}(\lambda)); from condition (13) it follows that (|a(\lambda)|^2\leq M) almost everywhere.

The considerations stated make it possible to conclude that the elements of the matrix (\varphi(t,\tau,\lambda)) in the case of a Hilbert system ({x_k(t)}) are square-integrable.

Next, to the quantities (\hat{x}_k(t,T)) of best interpolation of the unknown values (x_k(t)), (k=1,\ldots,n), (t\in T), from the values (x_k(s)), (k=1,\ldots,n), (s\notin T), observed at the remaining instants of time, there correspond in the space (L^2) vector-functions whose components have the form (5)

[
e^{i\lambda t}\delta_k-\sum_{j=1}^{n}p_{kj}(\lambda)\sum_{s\in T}c_s e^{i\lambda s},
\tag{16}
]

where (p_{kj}(\lambda)) are the elements of the matrix (f^{-1}(\lambda)).

Summarizing what has been said, we obtain Theorem 4.

Theorem 4. If the system ({x_k(t)}) is Hilbert, then the quantities (\hat{x}_k(t,\tau)) of best extrapolation of the unknown values (x_k(t)), (k=1,\ldots,n), from the known values (x_k(s)), (k=1,\ldots,n), (s\leq\tau), are expressible in the form of the series (4) in these values.

If, in addition, the condition

[
\int_{-\pi}^{\pi}\frac{1}{[\operatorname{Sp} f(\lambda)]^2}\,d\lambda<\infty,
\tag{17}
]

is satisfied,

then, in the form of the series (4), they are expressed through the known values (x_k(s)), (k=1,\ldots,n), (s\in T), and the quantities (\hat x_k(t,T)) of the best interpolation of the unknown values (x_k(t)), (k=1,\ldots,n), (t\in T).

Following (4), we shall say that the system ({x_k(t)}) forms an unconditional basis if, under any permutation of the quantities (x_k(t)), it is a basis.*

As I. M. Gelfand showed (4), the system ({x_k(t)}) forms an unconditional basis if and only if it is simultaneously both a Bessel and a Hilbert system.

Combining the results obtained above, we obtain Theorem 5.

Theorem 5. In order that the system ({x_k(t)}) be an unconditional basis, it is necessary and sufficient that the condition

[
mI \leqslant f(\lambda) \leqslant MI
\tag{18}
]

hold for some (m>0), (M<\infty), for almost all (\lambda).

V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
19 X 1959

REFERENCES

¹ A. N. Kolmogorov, Bull. Moscow State Univ., 2, No. 6, 1 (1941).
² Yu. A. Rozanov, DAN, 116, No. 6, 22 (1957).
³ N. K. Bari, Uch. zap. MGU, 4, issue 148, 69 (1951).
⁴ I. M. Gelfand, Uch. zap. MGU, 4, issue 148, 224 (1951).
⁵ Yu. A. Rozanov, Uspekhi Mat. Nauk, 13, issue 2 (80), 93 (1958).
⁶ N. Wiener, Acta Math., 98, No. 1, 2, 111 (1957); P. Masani, Acta Math., 99, No. 1, 2, 93 (1958).
⁷ P. Masani, C. R., 246, 15, 2215 (1958).
⁸ P. Masani, C. R., 246, 2337 (1958).

* That is, the sum of the series in (4) does not change under a permutation of the terms.