# Wold's decomposition theorem for stationary processes

The posts

How come the deterministic part of Wold decomposition does not violate stationarity?

More about the deterministic part of Wold decomposition

express some concerns as regards the "deterministic" component in the Wold decomposition of a stationary stochastic process.

Comments and answers there (including mine), make assertions and provide some input, but appear incomplete.

So here are the questions

Q1: In what sense one of the components in the Wold decomposition is "deterministic"?

Q2: Is this component stationary?

See you on the other side.

The original source is the book Wold H. (1938) A study in the analysis of stationary time series.

Wold examines exclusively "discrete" stochastic processes, where the index of the process is discrete.

As regards the term "stationary" Wold appears to use what is today called "strict stationarity", since in Introduction, p. 4 he writes

When expected values appear, it is understood that they exist.

As one more preparatory step, in Chapter II, Section 14. p.41 he defines a "singular" distribution as follows

$$P[\cdot ]$$ is Probability, the alphas are real coefficients. As regards the $$m$$-terms, he sets them equal to the unconditional expected value of each r.v. "without loss of generality".

He then transfers the singularity concept to a stationary stochastic process by implicitly setting all $$m$$ equal to the unconditional expected value of the process, by writing its defining property as (p. 44)

and then "when con-

One can suspect why "singular" became "deterministic" in modern times: If we re-arrange eq. $$(77)$$ above, the variable $$\xi(t)$$ is exactly determined by its past, it is a "deterministic function" of its past.

The Wold decomposition

To my understanding the Theorem that includes the famous Wold decomposition is to be found in Chapter II, Section 20 "A canonical form of the discrete stationary process", p. 89, as Theorem 7, which goes

NOTATION ALERT: What nowadays we denote by $$\varepsilon(t)$$ or $$u(t)$$, Wold denotes by $$\eta(t)$$; and what we denote by $$\eta(t)$$, Wold denotes by $$\psi(t)$$. So in Wold's notation. $$\zeta(t)$$ is the familiar MA process, and $$\psi(t)$$ is the "deterministic" part, and hence our main object of interest.

Note 1: It is clear from the wording of the theorem, that $$\psi(t)$$ is a stationary process.

Note 2: Given the preliminaries, $$\psi(t)$$ satisfies exactly a linear relationship with respect to its own past, and not with respect to the past of the "whole" process $$\xi(t)$$.

Note 3: Nowhere does Wold require the whole "infinite" past for the singularity property.

But various references state that $$\psi(t)$$ can be predicted perfectly with respect to the past values of $$\xi(t)$$ (negating Note 2), and/or that we need the infinite past for that (negating Note 3).

It appears that it is Anderson T.W. (1971), The Statistical Analysis of Time Series, Ch. 7, Section 7.6.3. pp. 420-421, who is the one that cemented the transfer from "singular" to "deterministic", while respecting both Note 2 and Note 3 above. Using $$v_t$$ for Wold's $$\psi(t)$$, Anderson writes

Earlier he has defined the term "deterministic" as "perfect prediction from the past values". Note also that he writes "(possibly infinite) linear combination" (so not necessarily infinite), and he states clearly that we refer to the own past of the deterministic/singular component.