Exact Computation of Non-Gaussian Mismatch Penalties in Wiener-Hermite Cross-Correlation Identification
TL;DR
The computation is a weighted-$L^2$ projection whose core normal-system correspondence is machine-checked in Lean 4.5, and gives the exact finite-order excess $L^2(P)$ risk of this mismatch.
Abstract
Wiener-Hermite cross-correlation identification represents a polynomial response in the Hermite basis. Under Gaussian excitation the basis is orthogonal and a diagonal rule recovers it exactly; under non-Gaussian excitation the same basis is kept, but its Gram matrix gains off-diagonal terms and the diagonal rule is no longer the population projection. We give the exact finite-order excess $L^2(P)$ risk of this mismatch: a moment quadratic form from two Hankel-Cholesky factorizations and one diagonal solve, at $O(s^3)$ cost from moments to order $2s$. Closed cumulant forms at orders three and four expose which non-Gaussian features drive it; symmetry protects the Gaussian basis only through order two. A bootstrap decides, from data, whether a matched basis is worth building; on a Wiener-Hammerstein benchmark it separates a near-Gaussian channel (penalty $\approx 10^{-4}$) from a skewed output (penalty $0.05$). The computation is a weighted-$L^2$ projection whose core normal-system correspondence is machine-checked in Lean 4.