63. Averaging transformations has a covariance defect
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A common discretization replaces a family of matrices \(K(y)\) by its mean and then applies \(X\mapsto KXK^{\mathsf T}\). This is generally the wrong mean operation.
Proposition 57: Exact covariance correction for an averaged congruence
Let \(\nu\) be a probability measure and suppose \(K(y)\) has square-integrable entries. Put \(\bar K=\int K(y)\,\nu(dy)\). For every symmetric \(X\),
The correction is positive semidefinite when \(X\succeq0\). For an affine matrix function \(K(u)=K_0+uK_1\) with scalar \(u\) of finite second moment, it equals \(\operatorname{Var}_\nu(u)K_1XK_1^{\mathsf T}\). On a uniform interval of width \(h\), the variance is \(h^2/12\).
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Proof
Substitute \(K=\bar K+(K-\bar K)\) and expand. The two cross terms integrate to zero. The remaining integrand is a positive congruence when \(X\) is positive. For an affine matrix function the centered term is \((u-\mathbb E_\nu u)K_1\), giving the variance formula. Integrating the square of a centered uniform variable gives \(h^2/12\). \(\square\)
This is a defect between two operations: averaging a transformation parameter and applying its quadratic action. It is not the same as the residual from dropping a signal packet, although both fit the framework's requirement to record failures of commutation.
For the example in Section 66, equal bins with midpoint evaluation lose total mass
when there are \(n\) bins. With one bin, midpoint evaluation retains only one quarter of the required mass. Exact integration or the covariance correction restores the missing contribution. Dividing an inaccurate approximation by a guessed normalizer would define another model rather than establish this identity.