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(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.