MATHEMATICS

TSIAN TSE-PEI

ON LINEAR EXTRAPOLATION OF A DISCRETE HOMOGENEOUS RANDOM FIELD

(Presented by Academician A. N. Kolmogorov on 20 VIII 1956)

Definition. A discrete homogeneous random field is a family of complex random variables \(x(s,t)\), where \(s,t\) are integers and \(-\infty < s < \infty\), \(-\infty < t < \infty\), such that \(M|x(s,t)|^2 < \infty\) and

\[ B_x(s,t)=M[x(s+m,t+n)\overline{x(s+t)}] \tag{1} \]

does not depend on \(m\) and \(n\).

The function \(B_x(s,t)\) is, obviously, positive definite and therefore (see, for example, \(\left({}^{1}\right)\)):

\[ B_x(s,t)=\int_{-\pi}^{\pi}\int_{-\pi}^{\pi} e^{i(s\lambda+t\mu)}\,dF_x(\lambda,\mu), \tag{2} \]

where \(F_x(\lambda,\mu)\) is an unnormalized two-dimensional distribution function, called the spectral function of the field \(\{x(s,t)\}\).

Let \(H_x\) be the minimal closed linear subspace containing all the variables \(x(s,t)\), and let \(H_x(t)\) be the minimal closed linear subspace containing all \(x(m,n)\) with \(-\infty < m < \infty\) and \(n \le t\). Denote

\[ S_x=\bigcap_t H_x(t). \tag{3} \]

If

\[ S_x=H_x, \tag{4} \]

then the homogeneous random field \(\{x(s,t)\}\) will be called singular.

Every element \(x(s,t)\) of the field \(\{x(s,t)\}\) is uniquely represented in the form of the sum

\[ x(s,t)=\eta(s,t)+\xi(s,t), \tag{5} \]

where \(\xi(s,t)\in H_x(0)\), and \(\eta(s,t)\) is orthogonal to \(H_x(0)\).

Put

\[ \rho(s,t)=\|\eta(s,t)\|^2. \tag{6} \]

Obviously,

\[ \rho(s,t)=\rho(s',t)=\rho(t), \]

\[ \rho(t_1)\leq \rho(t_2)\quad \text{for } t_1\leq t_2, \tag{7} \]

so that there exists

\[ \lim_{t\to+\infty}\rho(t)=\sigma_\infty^2. \tag{8} \]

If

\[ \sigma_\infty^2=M|x(s,t)|^2=\|x\|^2, \tag{9} \]

then the homogeneous random field \(\{x(s,t)\}\) will be called regular (cf. \(\left({}^{1-3}\right)\)).

It is not difficult to see that, in order for the homogeneous random field \(\{x(s,t)\}\) to be regular, it is necessary and sufficient that the equality \(S_x=0\) hold.

We shall indicate the conditions imposed on the spectral function which guarantee the regularity or singularity of the field \(\{x(s,t)\}\).

Theorem 1. In order that the homogeneous random field \(\{x(s,t)\}\) be regular, it is necessary and sufficient that the following conditions be satisfied:

a) the measures \(dF_x(\lambda,\mu)\) and \(dF_x(\lambda,\pi)\,d\mu\) (i.e., the product of the measure \(dF_x(\lambda,\pi)\) by the measure \(d\mu\)) be absolutely continuous with respect to each other on the square \(-\pi \leq \lambda \leq \pi,\ -\pi \leq \mu \leq \pi\);

b) for almost all values of \(\lambda\) (with respect to the measure \(dF_x(\lambda,\pi)\)),

\[ \left| \int_{-\pi}^{\pi} \log\left( \frac{dF_x(\lambda,\mu)} {dF_x(\lambda,\pi)\,d\mu} \right)d\mu \right|<+\infty . \tag{10} \]

Theorem 2. In order that the homogeneous random field \(\{x(s,t)\}\) be regular, it is necessary and sufficient that its spectral function can be represented in the form

\[ F_x(\lambda,\mu)= \int_{-\pi}^{\lambda}\int_{-\pi}^{\mu} |L(\lambda,\mu)|^2\,dG(\lambda)\,d\mu, \tag{11} \]

where \(G(\lambda)\) is some function, nondecreasing on the interval \([-\pi,\pi]\) and such that \(G(\pi)-G(-\pi)>0\), and \(L(\lambda,\mu)\) is a complex-valued function, different from zero almost everywhere with respect to \(dG(\lambda)\,d\mu\), representable in the form

\[ L(\lambda,\mu)=\sum_{n=0}^{+\infty} l_n(\lambda)e^{-in\mu} \qquad \bigl(l_n(\lambda)\in L^2(dG(\lambda)),\ n=0,1,2,\ldots\bigr). \]

Theorems 1 and 2 give two forms of the necessary and sufficient condition for regularity. Let us note that the spectral function of a regular homogeneous random field need not be absolutely continuous, as the following simple example shows:

\[ x(s,t)\equiv x(t), \]

where \(x(t)\) is a regular stationary random sequence.

Theorem 3. In order that the spectral function of a regular homogeneous random field \(\{x(s,t)\}\) be absolutely continuous, it is necessary and sufficient that \(x(s,t)\) can be represented in the form

\[ x(s,t)=\sum_{n=0}^{+\infty}\sum_{m=-\infty}^{+\infty} a_{mn}u(s-m,t-n), \tag{12} \]

where \(\{u(s,t)\}\) is a family of pairwise uncorrelated random variables with constant variance.

Theorem 4. In order that the homogeneous random field \(\{x(s,t)\}\) be singular, it is necessary and sufficient that on the interval \(-\pi \leq \lambda \leq \pi\) the condition

\[ \int_{-\pi}^{\pi} \log\left( \frac{dF_x(\lambda,\mu)} {dF_x(\lambda,\pi)\,d\mu} \right)d\mu=-\infty \tag{13} \]

hold almost everywhere (with respect to the measure \(dF_x(\lambda,\pi)\)), where

\[ \frac{dF_x(\lambda,\mu)}{dF_x(\lambda,\pi)\,d\mu} \]

is the absolutely continuous part of the measure \(dF_x(\lambda,\mu)\) with respect to the measure \(dF_x(\lambda,\pi)\,d\mu\).

We shall assume that \(Mx(s,t)=0\). Let

\[ x(s,m)=\beta(s,m)+\gamma(s,m), \tag{14} \]

where \(\gamma(s,m)=\operatorname{proj}_{H_x(-1)} x(s,m)\), and \(\beta(s,m)\) is orthogonal to \(H_x(-1)\). Denote

\[ \sigma_m^2=\|\beta(s,m)\|^2 . \tag{15} \]

(It is clear that this quantity does not depend on \(s\).) Then \(\sigma_m\) will be the mean-square error of linear extrapolation of the field \(\{x(s,t)\}\) \(m+1\) steps ahead with respect to the variable \(t\). In applications such a problem naturally arises in cases where \(t\) plays the role of a time variable, and \(s\) of a spatial one.

Theorem 5.

\[ \sigma_m^2=2\pi\int_{\eta_\lambda}\sum_{k=0}^{m}|\varphi_k(\lambda)|^2\,dF_x(\lambda,\pi), \tag{16} \]

where

\[ \eta_\lambda=\left\{\lambda;\ \left|\int_{-\pi}^{\pi}\log\left(\frac{dF_x(\lambda,\mu)}{dF_x(\lambda,\pi)\,d\mu}\right)d\mu\right|<+\infty\right\}, \tag{17} \]

and the functions \(\varphi_k(\lambda)\), \((\lambda\in\eta_\lambda)\), are determined from the relations

\[ \exp\left[\frac12 A_0(\lambda)+\sum_{k=1}^{+\infty}A_k(\lambda)\zeta^k\right] =\varphi_0(\lambda)+\varphi_1(\lambda)\zeta+\varphi_2(\lambda)\zeta^2+\cdots; \tag{18} \]

\[ A_k(\lambda)=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{ik\mu}\log\left(\frac{dF_x(\lambda,\mu)}{dF_x(\lambda,\pi)\,d\mu}\right)d\mu . \tag{19} \]

In particular,

\[ \sigma_0^2=2\pi\int_{-\pi}^{\pi} \exp\left\{\frac{1}{2\pi}\int_{-\pi}^{\pi} \log\left(\frac{dF_x(\lambda,\mu)}{dF_x(\lambda,\pi)\,d\mu}\right)d\mu\right\} dF_x(\lambda,\pi). \tag{20} \]

Formulas (16), (20) are analogous to Kolmogorov’s formulas \((2,3)\) for the mean square of the error of extrapolation of stationary random sequences.

Definition. A homogeneous random field \(\{x(s,t)\}\) is called a field of Markov type if \(\operatorname{proj}_{H_x(t-1)} x(s,t)\) belongs to the closed linear manifold spanned by the quantities \(x(s,t-1)\), \(-\infty<s<+\infty\).

Theorem 6. I. In order that a homogeneous random field \(\{x(s,t)\}\) be a field of Markov type, it is sufficient that the condition

\[ F_x(\omega)=\iint_{\omega}\frac{1}{|1-l(\lambda)e^{-i\mu}|^2}\,dG(\lambda)\,d\mu \quad\text{for all }\omega\subseteq\Omega, \tag{21} \]

be satisfied, where:

a) \(G(\lambda)\) is some real nondecreasing bounded function on the interval \(-\pi\leq\lambda\leq\pi\), with \(G(\pi)-G(-\pi)>0\);

b) \(l(\lambda)\) is a complex function such that \(|l(\lambda)|\leq 1\) and

\[ \int_{-\pi}^{\pi}\frac{dG(\lambda)}{1-|l(\lambda)|^2}<+\infty; \tag{22} \]

c)

\[ \Omega=\{(\lambda,\mu);\ 1-l(\lambda)e^{-i\mu}\ne0\}. \tag{23} \]

II. If the field \(\{x(s,t)\}\) is a field of Markov type, then there exist functions \(l(\lambda)\) and \(q(\lambda)\) such that

\[ F_x(\omega)=\iint_{\omega}\frac{q(\lambda)}{|1-l(\lambda)e^{-i\mu}|^2}\,dF_x(\lambda,\pi)\,d\mu \tag{24} \]

for all \(\omega \subseteq \Omega\), where

\[ \text{a)}\quad \Omega=\{(\lambda,\mu);\;1-l(\lambda)e^{-i\mu}\ne 0\}; \]

\[ \text{b)}\quad |l(\lambda)|\leq 1 \qquad (-\pi\leq \lambda\leq \pi); \]

\[ \text{c)}\quad q(\lambda)\geq 0 \qquad (-\pi\leq \lambda\leq \pi). \]

III. The field \(\{x(s,t)\}\) is singular if and only if \(q(\lambda)\equiv 0\), and is regular if and only if \(F_x(\overline{\Omega})=0\), where \(\overline{\Omega}\) is the complement of \(\Omega\).

Theorem 7. If the homogeneous random field \(\{x(s,t)\}\) is a regular field of Markov type, then

\[ \operatorname{proj}_{H_{x(t-1)}} x(s,t) = \int_{-\pi}^{\pi}\int_{-\pi}^{\pi} e^{i[s\lambda+(t-1)\mu]} l(\lambda)\,dr_x(\lambda,\mu), \tag{25} \]

where

\[ l(\lambda)= \int_{-\pi}^{\pi} e^{i\mu}\, \frac{dF_x(\lambda,\mu)} {dF_x(\lambda,\pi)\,d\mu} \,d\mu . \]

The proofs of Theorems 1–3 and 5 are carried out analogously to the proofs of the corresponding theorems for the one-dimensional case (see \((^2,^3)\), and also \((^{4-6})\)).

The author is grateful to A. M. Yaglom for posing the problems considered in the present note.

Received
20 VIII 1956

REFERENCES

\(^1\) A. M. Yaglom, Uspekhi Mat. Nauk, 7, issue 5 (51), 3 (1952).
\(^2\) A. N. Kolmogorov, Byull. MGU, 2, issue 6, 1 (1941).
\(^3\) A. N. Kolmogorov, Izv. AN SSSR, Ser. Mat., 5, No. 1, 3 (1941).
\(^4\) F. Karhunen, Ann. Acad. Sci. Fennicae, A 1, No. 37, 3 (1947).
\(^5\) N. I. Akhiezer, Lectures on the Theory of Approximation, Moscow–Leningrad, 1947.
\(^6\) J. L. Doob, Stochastic Processes, N. Y., 1953.