TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Existence and uniqueness of the finite-history predictor class

Statement

Let $X = (X_{t})_{t \in Z}$ be a weakly stationary real process and let $n, h \geq 1$ . The affine $L^{2}$ minimization defining the finite-history best linear forecast of $X_{n + h}$ from $X_{1}, \dots, X_{n}$ has a unique minimizing almost-sure class, even when the history covariance matrix is singular.

Facts & Assumptions

Given: A weakly stationary real process $X$ on one probability space and integers $n, h \geq 1$ .

[F1]

Weak stationarity gives square-integrability of every $X_{t}$ . Strict and weak stationarity for integer-indexed time series

[F2]

For finitely many square-integrable real variables on one probability space, the affine $L^{2}$ projection has a unique minimizing class and an affine representative; singular covariance may make coefficients nonunique, but any two coefficient solutions yield the same predictor class. Normal equations for best affine $L^{2}$ prediction

[F3]

The finite-history forecast minimizes squared $L^{2}$ error over the affine span of the constant and $X_{1}, \dots, X_{n}$ classes. Best linear forecast from a finite history

Proof

1.1givenF1F2

Fix $n, h \geq 1$ and take target $X_{n + h}$ with regressors $X_{1}, \dots, X_{n}$ . By [F1] these variables are square-integrable, so [F2] supplies a unique minimizing almost-sure class in their affine span.

2.1F2F3step 1.1∎

By [F3], the affine span used in [F2] is exactly $M_{n}$ , so the unique minimizing class from step 1.1 is the finite-history forecast. The uniqueness asserted is of the $L^{2}$ class, not necessarily of its coefficients; this remains true when the history covariance is singular by [F2].

Depends on

Used by

Cited to discharge well-definedness by Best linear forecast from a finite history.

Dependency tree · two levels

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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)