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QUESTION-248: In a stable linear Gaussian VAR(1), does a scalar linear coarse variable whose reduced dynamics carry memory admit an exact finite-dimensional Markovian augmentation?

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Tagscomplex-systems, mathematics
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RefinesQUESTION-222 — Beyond the scalar linear Gaussian case and the deterministic case of a fast, contracting eliminated variable, when a coarse variable's reduced dynamics carry memory of the eliminated variables, does it admit a finite-dimensional Markovian augmentation whose prediction error is bounded in terms of the eliminated dynamics, and is a level whose history, not its present state, does the predicting still a level?
Answered byCLAIM-398 — In a stable linear Gaussian VAR(1) on ℝⁿ, a scalar linear coarse variable Y = LX is, in its own law, an ARMA process of orders at most (n, n − 1); so its Mori–Zwanzig memory, though non-zero at every lag in general, is carried exactly by finitely many coordinates that are functions of Y's own past and innovations

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Answer

Yes. CLAIM-398 (Active, demonstrated): in a stable linear Gaussian VAR(1) on ℝⁿ, a scalar linear coarse variable Y = LX is, in its own law, an ARMA process of orders at most (n, n − 1). Its Mori–Zwanzig memory is non-zero at every lag in general. Still, finitely many coordinates, all functions of Y’s own past and innovations, carry it exactly, so the prediction error of the augmentation is that of the optimal predictor, with nothing added. The augmentation is the one THEORY-257’s criterion conditions on.

CLAIM-397 adds that, for any scalar stationary process, the memory kernels are fixed by the autocovariance alone. So an own-history representation exists in second order for every stationary variable, and having one marks no level.

What it leaves to QUESTION-222

  • Other classes. Nonlinear dynamics, noisy or slow eliminated variables, and vector coarse variables. THEORY-762 and THEORY-773 cover two further cases.
  • Error in terms of the eliminated spectrum. In CLAIM-398’s one example, the cost of truncating to AR(p) fell at a rate set by the moving-average zero, not by the spectrum of the eliminated dynamics.
  • The ruling on whether the record’s level criteria apply to the augmented state or to the instantaneous one.

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