Source
Barnett and Seth (2023), Physical Review E 108, 014304, LIT-979, read in full in NOTE-791 (arXiv:2106.06511v2): the definition of dynamical independence (2.3)–(2.6), the VAR criterion (3.26) and the genericity remark of §3.3. The criterion is the paper’s. The derivation, the measure argument and the p = 1 counterexample below are the record’s.
What was actually shown
Model: X_t = Σ_{k=1}^p A_k X_{t−k} + ε_t in ℝⁿ, stable, with ε_t i.i.d. N(0, Σ), Σ ≻ 0. A coarse-graining Y = LX, with L an m × n matrix of full row rank, 0 < m < n, is dynamically independent when the transfer entropy T(X → Y) = I(Y_t : X⁻ | Y⁻) vanishes. Let W be L’s row space.
Sufficiency. If L A_k = B_k L for every k, equivalently W·A_k ⊆ W, then Y_t − Σ_k B_k Y_{t−k} = Lε_t, which is independent of the whole past of X. So T(X → Y) = 0.
Necessity. With Gaussian laws, T(X → Y) = 0 means E[Y_t | X⁻] = E[Y_t | Y⁻]. Write X_s = Σ_{j≥0} Ψ_j ε_{s−j}, with Ψ_0 = I and Ψ_j = Σ_{k=1}^{min(j,p)} A_k Ψ_{j−k}. Then E[Y_t | X⁻] = Σ_{j≥1} LΨ_j ε_{t−j}. In an element Z of the closed span of {Y_s : s < t}, the coefficient of ε_{t−j} is Cov(Z, ε_{t−j})Σ⁻¹. For a finite combination Σ_s C_s Y_{t−s} this coefficient is Σ_{s=1}^{j} C_s LΨ_{j−s}. For each j ≤ p these tuples form a finite-dimensional, hence closed, set, so limits keep the form. So LΨ_j = Σ_{s=1}^{j} C_s LΨ_{j−s} for j = 1, …, p. At j = 1 this is L A_1 = C_1 L. If L A_i = B_i L for all i < j, then LΨ_i = D_i L for i < j, and the j-th equation becomes L A_j + (something)·L = (something)·L. So L A_j = E_j L. By induction, the condition holds for every lag. This is (3.26), L A_k Mᵀ = 0 for k = 1, …, p, with the rows of M spanning ker L.
p ≥ 2: almost never. For a fixed W in the Grassmannian Gr(m, n), the map A ↦ L A Mᵀ is onto ℝ^{m×(n−m)}. So each lag’s condition is m(n − m) independent linear equations, and all p together are p·m(n − m). The set of pairs ((A_k), W) meeting them is a vector bundle over Gr(m, n) of total dimension pn² − (p − 1)m(n − m). That is less than pn² when p ≥ 2. Its projection to the parameter space ℝ^{pn²} is smooth, so its image is Lebesgue-null. A finite union over m keeps it null, and stability is an open condition. So almost every stable VAR(p) with p ≥ 2 has no exactly dynamically independent linear coarse-graining at any scale.
p = 1: open sets, not every scale. Exact independence at scale m holds iff W is an m-dimensional subspace invariant under Aᵀ. Such subspaces exist for all m on an open set, for example when every eigenvalue is real and distinct. But A = ½·[[0, −1], [1, 0]] is stable with eigenvalues ±i/2. Every matrix near it has non-real eigenvalues, and so no real one-dimensional invariant subspace. So on an open set of stable VAR(1) models in ℝ², no scale-1 coarse-graining is exactly independent. In general, a scale m fails whenever Aᵀ has no real invariant subspace of dimension m.
What could have failed: a generic VAR(p) could have had exact solutions from nonlinear coincidences between lags. It cannot, because the conditions are linear in the parameters for each fixed subspace.
What this does not say
- Not that near-independent variables are rare. It concerns exact zeros. The paper’s “emergence portrait” is built from near-independent optima, which this result leaves open.
- Not about nonlinear coarse-grainings or non-Gaussian noise. The criterion uses linear maps and Gaussian conditional laws.
- Not about state-space models with r > 1. The paper’s (3.25) and its stated ranges of m were not re-derived.
- Not that the dynamics fix the variables. That reading belongs to THEORY-257 and stays Proposed.