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

\[ \hat{\varphi}_k(t,\tau,\lambda)=\{\hat{\varphi}_{k1}(\lambda),\ldots,\hat{\varphi}_{kn}(\lambda)\}, \]

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.