MATHEMATICS

E. G. GLADYSHEV

ON PERIODICALLY CORRELATED RANDOM SEQUENCES

(Presented by Academician A. N. Kolmogorov, 22 X 1960)

1. Definition. A random sequence \(x_n\), \(n=0,\pm 1,\ldots\), is called periodically correlated if \(\mathbf{M}|x_n|^2<\infty\) for all \(n\) and there exists an integer \(T\) such that, for all \(n\) and \(\tau\),

\[ \mathbf{M}x_n=\mathbf{M}x_{n+T},\qquad \mathbf{M}x_{n+\tau}\overline{x}_n=\mathbf{M}x_{n+T+\tau}\overline{x}_{n+T}. \tag{1} \]

The number \(T\) in this case is called the period of the sequence \(x_n\).

In the problems considered below one may, without loss of generality, always assume that \(\mathbf{M}x_n=0\). In that case the principal characteristic of the sequence \(x_n\) will be its correlation function

\[ B(n,\tau)=\mathbf{M}x_{n+\tau}\overline{x}_n, \tag{2} \]

which is periodic in \(n\) with period \(T\) and therefore admits the representation

\[ B(n,\tau)=\sum_{k=0}^{T-1} B_k(\tau)\exp\left(\frac{2\pi i k n}{T}\right). \tag{3} \]

For convenience, extend the functions \(B_k(\tau)\), \(k=0,\ldots,T-1\), to all integer \(k\) by means of the equality \(B_k(\tau)=B_{k+T}(\tau)\).

Theorem 1. The function (3) will be the correlation function of some periodically correlated random sequence if and only if the matrix

\[ B(\tau)=\|B_{jk}(\tau)\|_{j,k=0,\ldots,T-1}, \tag{4} \]

where

\[ B_{jk}(\tau)=B_{k-j}(\tau)\exp(2\pi i j/T), \tag{5} \]

is the correlation matrix of some \(T\)-dimensional stationary random sequence.

Proof. We have to show that the validity, for arbitrary integers \(t_1,t_2,\ldots,t_n\) and arbitrary complex \(a_1,a_2,\ldots,a_n\), of the inequality

\[ \sum_{p,q=1}^{n} B(t_q,t_p-t_q)a_p\overline{a}_q \ge 0, \tag{6} \]

where \(B(t,\tau)\) is the function (3), is equivalent to the validity, for arbitrary \(k_1,\ldots,k_n\) taking integer values from \(0\) to \(T-1\), and for the same \(t_1,\ldots,t_n,a_1,\ldots,a_n\) as before, of the inequality

\[ \sum_{p,q=1}^{n}\{B_{k_p k_q}(t_p-t_q)\}a_p\overline{a}_q \ge 0, \tag{7} \]

where \(B_{jk}(\tau)\) are the functions (5). Suppose first that (6) holds. In this case, using the fact that, by virtue of (3),

\[ B_l(\tau)=\frac{1}{T}\sum_{n=0}^{T-1} B(n,\tau)\exp\left(-\frac{2\pi i n l}{T}\right), \tag{8} \]

we shall have:

\[ \sum_{p,q} B_{k_p k_q}(t_p-t_q)a_p\overline{a_q} = \sum_{p,q} B_{k_q-k_p}(t_p-t_q)\exp\left(\frac{2\pi i k_p(t_p-t_q)}{T}\right)a_p\overline{a_q} = \]

\[ = \frac{1}{T}\sum_{p,q}\exp\left(\frac{2\pi i k_p(t_p-t_q)}{T}\right)a_p\overline{a_q} \sum_{n=0}^{T-1} B(n,t_p-t_q)\exp\left(-\frac{2\pi i n(k_q-k_p)}{T}\right) = \]

\[ = \frac{1}{T}\sum_{m=0}^{T-1}\left[ \sum_{p,q} B(m+t_q,t_p-t_q)a_p \exp\left(\frac{2\pi i k_p(m+t_p)}{T}\right) \times \right. \]

\[ \left. \times\, \overline{ a_q \exp\left(\frac{2\pi i k_q(m+t_q)}{T}\right)} \right]\ge 0. \]

If, however, we assume that (7) is satisfied, then, by virtue of (5) and (3),

\[ \sum_{p,q} B(t_q,t_p-t_q)a_p\overline{a_q} = \sum_{p,q} a_p\overline{a_q}\sum_{l=0}^{T-1}B_l(t_p-t_q)\exp\left(\frac{2\pi i l t_q}{T}\right) = \]

\[ = \sum_{p,q} a_p\overline{a_q}\sum_{k=0}^{T-1}B_{k-j}(t_p-t_q)\exp\left(\frac{2\pi i t_q(k-j)}{T}\right) = \]

\[ = \frac{1}{T}\sum_{p,q}a_p\overline{a_q}\sum_{i,k}B_{k-j}(t_p-t_q) \exp\left(\frac{2\pi i j(t_p-t_q)}{T}\right) \exp\left(\frac{2\pi i(kt_q-jt_p)}{T}\right) = \]

\[ = \frac{1}{T}\sum_{p,q}a_p\overline{a_q}\sum_{j,k}B_{jk}(t_p-t_q) \exp\left(\frac{2\pi i(kt_q-jt_p)}{T}\right)\ge 0, \]

and thus Theorem 1 is proved.

1. Using now the spectral representation of the correlation matrix of a multidimensional stationary random sequence, we can formulate Theorem 1 also in the following form:

In order that the function (3) be the correlation function of some periodically correlated sequence, it is necessary and sufficient that the functions \(B_k(\tau)\), \(k=0,\ldots,T-1\), all be representable in the form of Fourier–Stieltjes integrals

\[ B_k(\tau)=\int_{0}^{2\pi} e^{i\tau\lambda}\,dF_k(\lambda), \tag{9} \]

where \(F_k(\lambda)\), \(k=0,\ldots,T-1\), are complex functions of bounded variation, continuous from the left, such that for any \(\lambda_1\) and \(\lambda_2\ge \lambda_1\) the increment \(\mathcal F(\lambda_2)-\mathcal F(\lambda_1)\) of the matrix

\[ \mathcal F(\lambda)=\left\|F_{k-j}((\lambda-2\pi j)/T)\right\|_{j,k=0,\ldots,T-1} \tag{10} \]

(for \(k<0\) and for \(\lambda<0\) or \(\lambda>2\pi\) the functions \(F_k(\lambda)\) are extended by means of the equalities \(F_k(\lambda)=F_{k+T}(\lambda)\), \(F_k(\lambda+2\pi)=F_k(\lambda)+F_k(2\pi)\), \(F_k(0)=0\) for all \(k\)) be a Hermitian non-negative definite matrix.

Let us also note that from formulas (3) and (9) it follows easily that the periodically correlated sequence \(x_n\) will necessarily be harmonizable in the sense of (1), i.e., will admit the representation

\[ x_n=\int_{0}^{2\pi} e^{in\lambda}z(d\lambda), \tag{11} \]

where \(z(\Lambda)\) is a random function of a set with zero mathematical expectation, satisfying the condition

\[ \mathbf{M} z(\Lambda)\overline{z(M)} = \sum_{k=-T+1}^{T-1} \int_{\Lambda \cap \left(M-\frac{2\pi k}{T}\right)} dF_k(\lambda) \tag{12} \]

(\(M-a\) is the set of points \(\mu-a,\ \mu\in M\)).

This can otherwise be expressed as follows: the spectrum of the random sequence \(x_n\) is concentrated on \(2T-1\) straight-line segments \(\lambda-\mu=2\pi k/T,\ k=-T+1,\ldots,T-1\), lying inside the square \(0\leqslant \lambda,\mu<2\pi\), and the functions \(F_k(\lambda)\) specify the spectral intensities on these \(2T-1\) lines. From this, of course, there also follows the condition formulated above concerning the matrix \(\mathfrak{F}(\lambda)\).

Formulas (11) and (12) also follow from the possibility of representing the periodically correlated sequence \(x_n\) in the form

\[ x_n=\sum_{k=0}^{T-1} z_n^k \exp\left(\frac{2\pi i k n}{T}\right), \tag{13} \]

where \(z_n=(z_n^0,\ldots,z_n^{T-1})\) is a \(T\)-dimensional stationary (in the broad sense) sequence given by the formula
\[ z_n^k=\frac{1}{T}\sum_{m=0}^{T-1} y_{\frac{n-m}{T}}^m \exp\left(-\frac{2\pi i m k}{T}\right), \]
and \(y_t=(y_t^0,\ldots,y_t^{T-1})\) is a \(T\)-dimensional random process obtained by a formal substitution, in the spectral expansion of the \(T\)-dimensional stationary sequence
\[ y_n=(x_{nT},x_{nT+1},\ldots,x_{nT+T-1}), \]
of the discrete parameter \(n\) by the continuous parameter \(t\).

Remark. The case of multidimensional periodically correlated random sequences
\[ x_n=(x_n^1,\ldots,x_n^s), \]
for which

\[ \mathbf{M}x_n^l=\mathbf{M}x_{n+T}^l,\qquad \mathbf{M}x_{n+\tau}^l \overline{x_n^m} = \mathbf{M}x_{n+T+\tau}^l \overline{x_{n+T}^m}. \tag{14} \]

can be treated analogously. Assuming here also that \(\mathbf{M}x_n^k\equiv 0\), we must take as the basic statistical characteristics of such a multidimensional sequence the correlation functions

\[ \mathbf{M}x_{n+\tau}^l \overline{x_n^m} = B^{lm}(n,\tau) = \sum_{k=0}^{T-1} B_k^{lm}(\tau) \exp\left(\frac{2\pi i k n}{T}\right). \tag{15} \]

In order that the functions (15) be the correlation functions of some multidimensional \(T\)-periodic random sequence, it is necessary and sufficient that the \((sT\times sT)\)-matrix
\[ B(\tau)=\|B_{lm}(\tau)\|_{l,m=1,\ldots,s}, \qquad B_{lm}(\tau)=\|B_{jk}^{lm}(\tau)\|_{j,k=0,\ldots,T-1}, \]
where
\[ B_{jk}^{lm}(\tau) = B_{k-j}^{lm}(\tau)\exp\left(\frac{2\pi i j\tau}{T}\right), \]
be the correlation matrix of some \(sT\)-dimensional stationary random sequence. Otherwise this can be formulated as follows: the functions \(B_k^{lm}(\tau)\) must be representable in the form
\[ B_k^{lm}(\tau)=\int_0^{2\pi} e^{i\tau\lambda}\,dF_k^{lm}(\lambda), \]
where the left-continuous complex functions of bounded variation \(F_k^{lm}(\lambda)\) possess the property that the \((sT\times sT)\)-matrix
\[ \mathfrak{F}(\lambda) = \|F_{lm}(\lambda)\|_{l,m=1,\ldots,s}, \qquad F_{lm}(\lambda) = \left\|F_{k-j}^{lm}\left(\frac{\varkappa-2\pi j}{T}\right)\right\|_{j,k=0,\ldots,T-1} \]
is a matrix with Hermitian nonnegative-definite increments.

The further results of the present paper can also without difficulty be carried over to the case of multidimensional periodically correlated sequences; however, for simplicity of notation we shall henceforth consider only the one-dimensional case.

1. It is easy to see that the random sequence \(x_n,\ n=0,\pm1,\ldots,\) will be periodically correlated with period \(T\) if and only if the \(T\)-dimensional random sequence \(y_n=(y_n^0,\ldots,y_n^{T-1})\), where \(y_n^k=x_{nT+k},\ -\infty<n<\infty,\ k=0,\ldots,T-1,\) is stationary in the wide sense, i.e.,
   \[ \mathbf{M}y_{n+\tau}^j \overline{y_n^k} = \int_0^{2\pi} e^{i\tau\lambda}\,dF_{jk}(\lambda), \qquad j,k=0,\ldots,T-1 . \tag{16} \]

Denote by \(H_x\), \(H_x(n)\), and \(\hat H_x(n)\) the linear closures (in the mean square), respectively, of the random variables \(\{x_\tau,\ -\infty<\tau<\infty\}\), \(\{x_\tau,\ \tau\leq n\}\), and \(\{x_\tau,\ \tau\ne n\}\). Following \((^2)\), we shall call a random sequence \(x_n\) regular if \(\bigcap_n H_x(n)=0\), and singular if \(\bigcap_n H_x(n)=H_x\). A regular sequence \(x_n\) for which, for all \(n\), \(x_n\notin H_n(n-1)\), will be called completely regular. We shall say that the sequence \(x_n\) is minimal if
\[ \min_{h\in \hat H_x(n)} \mathbf{M}|x_n-h|^2=\sigma_n^2>0 \]
for all \(n\). In this case the condition of complete regularity of the sequence \(x_n\) obviously coincides with the condition of regularity of highest rank (in the sense of \((^3)\)) of the multidimensional stationary sequence \(y_n\), while the condition of minimality of the sequence \(x_n\) coincides with the condition of minimality of highest rank (in the sense of \((^{4,5})\)) of the sequence \(y_n\).

Theorem 2. The matrix \(\left\|dF_{jk}(\lambda)/d\lambda\right\|_{j,k=0,\ldots,T-1}\) is determined by the equality
\[ \left\|\frac{dF_{jk}(\lambda)}{d\lambda}\right\|_{j,k=0,\ldots,T-1} = T U^{-1}(\lambda)\frac{d\mathcal{F}(\lambda)}{d\lambda}U(\lambda) \quad \text{for almost all } \lambda, \tag{17} \]
where \(d\mathcal{F}(\lambda)/d\lambda\) is the matrix of the derivatives of all matrix elements of the matrix \(\mathcal{F}(\lambda)\), and
\(U(\lambda)=\left\|U_{jk}(\lambda)\right\|_{j,k=0,\ldots,T-1}\) is a unitary matrix depending on \(\lambda\), with elements
\[ U_{jk}(\lambda)=T^{-1/2}\exp\left(\frac{2\pi ijk-ik\lambda}{T}\right). \]

From Theorem 2, by virtue of the results of \((^{3-5})\), the following propositions follow:

Theorem 3. A periodically correlated random sequence \(x_n\) is completely regular if and only if: 1) all \(F_l(\lambda)\) are absolutely continuous; 2) \(\det d\mathcal{F}(\lambda)/d\lambda>0\) almost everywhere; 3)
\[ \int_0^{2\pi}\ln \det \frac{d\mathcal{F}(\lambda)}{d\lambda}\,d\lambda>-\infty . \]

Theorem 4. A periodically correlated random sequence \(x_n\) is minimal if and only if: 1) \(\det d\mathcal{F}(\lambda)/d\lambda>0\) almost everywhere; 2)
\[ \int_0^{2\pi}\operatorname{Sp}\left(\frac{d\mathcal{F}(\lambda)}{d\lambda}\right)^{-1}d\lambda<\infty . \]

In conclusion, the author expresses his gratitude to A. M. Yaglom, whose valuable advice contributed to the improvement of the present note.

Institute of Atmospheric Physics
Academy of Sciences of the USSR

Received
10 X 1960

CITED LITERATURE

\({}^1\) A. Blanc-Lapierre, R. Fortet, Théorie des fonctions aléatoires, Paris, 1953.
\({}^2\) A. N. Kolmogorov, Bull. Moscow State Univ., 2, No. 6 (1941).
\({}^3\) V. N. Zasukhin, DAN, 33, 435 (1941).
\({}^4\) Yu. A. Rozanov, DAN, 116, No. 6, 22 (1957).
\({}^5\) P. Masani, C. R., 246, No. 15, 2215 (1958).