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A Gaussian finite-history forecast is a conditional expectation

Statement

Assume the Axiom of Choice. Let $X$ be a weakly stationary real process and let $n, h \geq 1$ . If the finite vector $(X_{n + h}, X_{n}, \dots, X_{1})$ is jointly multivariate normal, then the finite-history best linear forecast satisfies ${\hat{X}}_{n + h} = E [X_{n + h} ∣ σ (X_{1}, \dots, X_{n})]$ almost surely. The history covariance may be singular. Without joint Gaussianity the equality need not hold.

Facts & Assumptions

Given: AC, a weakly stationary real process $X$ , integers $n, h \geq 1$ , and a jointly multivariate-normal vector $(X_{n + h}, X_{n}, \dots, X_{1})$ .

[A1]

AC is inherited from the multivariate-normal and conditional-expectation suppliers used in this Gaussian branch. The Axiom of Choice

[F1]

The finite-history affine minimization has a unique predictor class even when the history covariance is singular. Existence and uniqueness of the finite-history predictor class

[F2]

The affine projection theorem supplies normal equations and an intercept for a minimizing affine representative, including with singular covariance. Normal equations for best affine $L^{2}$ prediction

[F3]

A multivariate-normal law is preserved by reading any finite linear projection as a scalar normal projection. Multivariate normal law, including singular covariance

[F4]

Jointly Gaussian blocks with zero cross-covariance are independent, with singular covariance allowed. Uncorrelated jointly Gaussian blocks are independent

[F5]

If an integrable random variable is independent of a sigma-algebra, its conditional expectation is its ordinary mean; a measurable integrable variable is its own conditional expectation. Conditioning a known variable and an independent variable

[F6]

Conditional expectation is linear on integrable variables and is unique as an almost-sure class. Basic algebra and order properties of conditional expectation Conditional expectation is unique almost surely

[F7]

Conditional expectation given a sigma-algebra is defined by measurability, integrability and equality of integrals on every event in that sigma-algebra. Conditional expectation given a sigma algebra

Proof

1.1givenF1F2

Let $Y = (X_{n}, \dots, X_{1})$ and choose an affine representative ${\hat{X}}_{n + h}$ of the unique class in [F1]. Put $R = X_{n + h} - {\hat{X}}_{n + h}$ . By [F2], its mean is zero by the intercept equation, and each normal equation gives $Cov (X_{n + h}, Y_{i}) = Cov ({\hat{X}}_{n + h}, Y_{i})$ , hence $Cov (R, Y_{i}) = 0$ .

2.1A1F3F4step 1.1

The vector $(R, Y)$ is an affine linear image of $(X_{n + h}, Y)$ , so every scalar linear projection is normal by [F3] and $(R, Y)$ is jointly multivariate normal. Step 1.1 gives zero cross-covariance between $R$ and each coordinate of $Y$ , so [F4] makes $R$ independent of $Y$ and therefore of $G : = σ (Y)$ .

3.1A1F5F6F7step 2.1∎

The forecast ${\hat{X}}_{n + h}$ is an affine function of $Y$ , hence $G$ -measurable and integrable. By [F5], $E [R ∣ G] = E [R] = 0$ and $E [{\hat{X}}_{n + h} ∣ G] = {\hat{X}}_{n + h}$ . Since $X_{n + h} = R + {\hat{X}}_{n + h}$ , [F6] makes the forecast a version of $E [X_{n + h} ∣ G]$ under [F7]; uniqueness in [F6] gives the almost-sure equality. AC is used only through the Gaussian and conditional-expectation branches.

Remarks

The companion page includes a finite three-state process where the conditional mean is not affine in the observed history.

Depends on

Used by

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Sources

Dewei Wang, STAT 720 Time Series Analysis, Section 5.2 (standard reference, not scraped)
Nicolai Meinshausen, Time Series Analysis: Notes for the course HS 2022, Section 2.7 (standard reference, not scraped)