4.2 Finite interaction and the strong-interaction boundary

Let \(P=\mathbf1\pi ^\top \) and \(N=W-P\). When \(P\) is an absorbing idempotent projection, the normalized Leontief multiplier has the exact complementary-resolvent remainder used in the next result.

Theorem 35 Finite-\(\rho \) resolvent bound

✓

If \(\lVert \rho (W-P)\rVert {\lt}1\), then

\[ \left\lVert (1-\rho )(I-\rho W)^{-1}-P\right\rVert \leq |1-\rho |\{ 1-\lVert \rho (W-P)\rVert \} ^{-1}\lVert I-P\rVert . \]

Proof ▼

Split the multiplier into the Perron projection and complementary resolvent, then apply the Neumann-series norm bound to the latter.

Theorem 36 Resolvent limit from the power limit

✓

For row-stochastic \(W\), the Perron power limit implies \((1-\rho )(I-\rho W)^{-1}\to P\) as \(\rho \uparrow 1\).

Proof ▼

The exact projection split leaves a complementary term multiplied by \(1-\rho \), which vanishes at the boundary.

Theorem 37 Rank-one collapse of rescaled covariance

✓

For innovation covariance \(V\), row-stochastic \(W\), and the Perron power limit,

\[ (1-\rho )^2\Sigma _{\mathrm{SAR}} \longrightarrow (\pi ^\top V\pi )\mathbf1\mathbf1^\top . \]

Proof ▼

Apply the normalized resolvent limit on both sides of the covariance congruence.