Proposition. Massive kernel factorization and its normalizer [fdip-0033]
✍️source

  • October 29, 2026

Let \(L\) have the kernel decomposition and let \(m>0\). The positive precision \(Q_m=L+m^2I\) has inverse and normalizer \[\begin {aligned} Q_m^{-1}&=m^{-2}P_K\\ &\quad +i(L_H+m^2I_H)^{-1}p_H,\\ Z_{Q_m}&=(\sqrt {2\pi }/m)^k Z_{H,m},\\ Z_{H,m}&=\frac {(2\pi )^{r/2}} {\sqrt {\prod _{\lambda _a>0}(\lambda _a+m^2)}}. \end {aligned}\] Under the addition isometry \(K\times H\to E\), its probability law is the product of the centered \(K\)-law with precision \(m^2I_K\) and the centered \(H\)-law with precision \(L_H+m^2I_H\). In particular its pushforward by \(p_H\) is exactly the second law.

Proof. 

  • October 29, 2026

On \(K\), \(Q_m=m^2I_K\); on \(H\), \(Q_m=L_H+m^2I_H\). Inverting these blocks gives the displayed inverse. The addition isometry preserves product volume, and the action splits: \[\begin {aligned} &\tfrac 12h(z+x,Q_m(z+x))\\ &\quad =\tfrac 12m^2\|z\|^2\\ &\qquad +\tfrac 12h(x,(L_H+m^2I_H)x). \end {aligned}\] Nonnegative Tonelli therefore factors the integral into the two normalizers. Positive normalization makes the first \((\sqrt {2\pi }/m)^k\) and the second the stated spectral product. Division by this product identifies the probability on every measurable rectangle with the product of the two probabilities, hence identifies the measures. Integrating out the \(K\) factor gives the projection law.

Every empty product is one. Thus these formulas hold when \(k=0\), \(r=0\), or \(E=0\). Since each positive eigenvalue stays positive, \[\begin {aligned} Z_{H,m}&\longrightarrow Z_H,\\ m^k Z_{Q_m}&\longrightarrow (2\pi )^{k/2}Z_H. \end {aligned}\] Here \(Z_H\) is the constrained normalizer. For \(k>0\), the full normalizer diverges. This is orthogonal kernel separation; it does not require a general block-elimination theorem.

This product calculation combines the orthogonal decomposition with the positive-covariance density of [friedli2017statistical, Proposition B.55, p. 505] after replacing covariance by inverse precision. The displayed kernel factor is derived here, including zero-dimensional factors.