Gaussian precision zeros characterize pairwise conditional independence

Fix distinct $i,j$. By [A1], [F1] and [F2], the joint density of $X$ is the stated nondegenerate Gaussian density; here positive definiteness is an additional hypothesis beyond the singular-covariance multivariate-normal definition.

Reorder coordinates so $Y=(X_i,X_j)$ and $Z=(X_k{)}_{k\mathrm{\notin }\{i,j\}}$. In the correspondingly permuted precision matrix write $\mathrm{\Theta }=\left(\begin{array}{cc}A& B\\ B^T& C\end{array}\right)$, where $A=\left(\begin{array}{cc}a& c\\ c& d\end{array}\right)$. For every nonzero $u\in {\mathbb{R}}^2$, $u^{T}Au=(u,0{)}^T\mathrm{\Theta }(u,0)>0$, so $A$ is positive definite. In particular, $a>0$ and evaluating the quadratic form of $A$ at $(-c\mathrm{/}a,1)$ gives $d-c^2\mathrm{/}a>0$, hence $\det A=ad-c^2>0$. For $p=2$, $Z$ ranges over the singleton ${\mathbb{R}}^0$ and the density of $Y$ is unconditional.

For $p>2$, [F7] supplies the countable-choice hypothesis of [F6], so the joint Euclidean density in [F2] is also a density relative to ${\lambda }_2\otimes {\lambda }_{p-2}$. Put $u=y-{\mu }_Y$, $w=z-{\mu }_Z$, and $D=C-B^T{A}^{-1}B$. For nonzero $w$, $(-A^{-1}Bw,w{)}^T\mathrm{\Theta }(-A^{-1}Bw,w)=w^{T}Dw>0$, hence $D$ is positive definite. Completing the square gives $u^{T}Au+2u^{T}Bw+w^{T}Cw=(u+A^{-1}Bw{)}^{T}A(u+A^{-1}Bw)+w^{T}Dw$. The section integral over $y$ is therefore positive and finite for every $z$: its $y$ factor is the normalized two-dimensional Gaussian density with precision $A$, multiplied by a finite positive constant.

For $p>2$, use the explicit filler probability concentrated at $(0,0)$ on ${\mathbb{R}}^2$ in [F3]; the positive finite marginal sections from step 3.1 show that the conditional law at every $z$ has density $\frac{\sqrt{\det A}}{2\pi }\exp (-\frac{1}{2}(u+A^{-1}Bw{)}^{T}A(u+A^{-1}Bw))$. It is $N_2(m(z),A^{-1})$, where $m(z)={\mu }_Y-A^{-1}B(z-{\mu }_Z)$. For $p=2$, conditioning on the constant $Z$ is unconditional; since $A=\mathrm{\Theta }$ in this case, the law is $N_2(m(\ast ),A^{-1})$ with $m(\ast )={\mu }_Y$. Thus in either case the conditional law is $N_2(m(z),A^{-1})$, interpreting $z=\ast$ when $p=2$.

If ${\mathrm{\Theta }}_{ij}=c=0$, then $A$ is diagonal. For $p>2$, the conditional density in step 4.1 is a product of two univariate Gaussian densities for every $z$; for $p=2$, the unconditional bivariate density is that product. For $p>2$, [F3] gives the conditional-event identities and [F9] identifies the corresponding conditional expectations; for $p=2$, conditional expectations given the constant $Z$ are ordinary expectations. In either case, simple-function approximation extends the indicator identities to bounded measurable functions. Tonelli's theorem [F10], applied to the positive and negative parts of the product integrand, factors the integrals against the product density. Hence the conditional expectations factor for all bounded measurable coordinate functions, and [F4] gives $X_i\perp \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\perp X_j\mid \sigma (Z)$.

Conversely, suppose $X_i\perp \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\perp X_j\mid \sigma (Z)$. Apply [F4] to the four bounded pairs formed from $\cos (X_i),\sin (X_i)$ and $\cos (X_j),\sin (X_j)$ and combine the resulting identities. For $p>2$, [F3] and [F9] identify these conditional expectations with integrals under the conditional law in step 4.1, so for almost every $z$ its joint characteristic function at $(1,1)$ equals the product of its conditional marginal characteristic functions. For $p=2$ the same identity is unconditional, at the unique value $z=\ast$. By [F1] and [F5], in either case the joint characteristic function is $\exp (i(m_1(z)+m_2(z))-\frac{1}{2}((A^{-1}{)}_{11}+2(A^{-1}{)}_{12}+(A^{-1}{)}_{22}))$, whereas the product of the marginals omits the middle covariance term. Equality implies $\exp (-(A^{-1}{)}_{12})=1$, and since $(A^{-1}{)}_{12}$ is real this forces $(A^{-1}{)}_{12}=0$.

The inverse of $A$ is $(ad-c^2{)}^{-1}\left(\begin{array}{cc}d& -c\\ -c& a\end{array}\right)$. Since $\det A>0$, $(A^{-1}{)}_{12}=0$ if and only if $c=0$, which is exactly ${\mathrm{\Theta }}_{ij}=0$. Steps 5.1 and 5.2 prove the two implications; by [F8], these are exactly the nonedge and conditional-independence cases of the Gaussian pairwise Markov graph.