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CLAIM-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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Properties8
StatusActive
Rolethesis
Warrantdemonstrated
Tagscomplex-systems, probabilistic-modeling, mathematics
Linedistributed-agency, formal-apparatus
AnswersQUESTION-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?
Rests onCLAIM-389 — In a stable linear Gaussian VAR(1), exact dynamical independence of a linear coarse-graining implies that its Mori–Zwanzig memory vanishes, but not conversely: memory needs coupling out of the coarse variable and back, and dependence needs only the way back, so a coarse variable can be Markov in its own law with zero memory and still be dynamically dependent
GroundsLIT-1304 — Discrete approach to stochastic parametrization and dimension reduction in nonlinear dynamics, THEORY-733 — For a stable linear Gaussian VAR(p), a linear coarse-graining is exactly dynamically independent iff its row space is invariant under the transpose of every lag matrix, so for p ≥ 2 almost no parameter set has one at any scale, while for p = 1 they are the transpose's invariant subspaces, which exist on open sets but not generically at every scale

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The claim

Take X_t = AX_{t−1} + ε_t, A stable, ε_t Gaussian white noise, and a scalar Y_t = LX_t. With B the backshift, (I − AB)X_t = ε_t. Multiplying by the adjugate gives det(I − AB)X_t = adj(I − AB)ε_t. det(I − AB) is a polynomial φ(B) of degree at most n, and each entry of adj(I − AB) is a polynomial of degree at most n − 1. So φ(B)Y_t = L adj(I − AB)ε_t is a finite moving combination of ε at lags 0 to n − 1. Its autocovariance vanishes beyond lag n − 1. A stationary Gaussian process with that property is an MA(q) with q ≤ n − 1 in its own innovations (spectral factorization), so Y is ARMA with orders at most (n, n − 1). Its state in innovations form consists of past Y and past innovations, which are functions of Y’s own past when the moving-average part is invertible.

Against CLAIM-389’s memory kernels M_j = K₁₂K₂₂^(j−1)K₂₁: these are non-zero at every lag in general, but by Cayley–Hamilton they satisfy a linear recurrence of order dim W⊥. The infinite memory is therefore finitely generated, and the ARMA form realizes it within Y’s own history.

Checks. On a random n = 3 model, φ(B)Y had autocovariances 1.643, 0.837 and 0.516 at lags 0–2 and below 10⁻¹⁶ at lags 3–6. For QUESTION-222’s own example (ẋ = −x + y, ẏ = 0.2x − 0.3y, with noise, sampled at h = 0.1), the continuous kernel 0.2e^(−0.3t) never vanishes, yet det(I − A_hB)x has autocovariance 0.1767 and −0.0879 at lags 0 and 1 and zero beyond: the sampled x is exactly ARMA(2, 1). Re-run by the coordinator.

What it does not say

  • Not the minimal order. That the ARMA order equals 1 plus the Hankel rank of the memory is not proved here. It is proved, for every scalar stationary process with rational autocovariance, in CLAIM-397. The ARMA state is a finite realization of the Wiener filter of THEORY-819.
  • Not a bound for truncation. Dropping the moving-average part leaves an AR(p) model whose extra error fell geometrically, at a rate set by the moving-average zero, in one example. That rate is neither an eigenvalue of A nor of K₂₂. It is numerical, not part of the claim.
  • Nothing beyond the linear Gaussian VAR(1). Chorin and Lu’s nonlinear closures (THEORY-818) are fitted and carry no such guarantee.
  • Not for vector Y without change. The same adjugate argument gives a VARMA, but its orders and identifiability are not checked here.

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