Finite Gaussian-vector distortion waterline

Since $1<\theta <4$, the water-level equation is $\min (4,\theta )+\min (1,\theta )=\theta +1=5\mathrm{/}2$, so $\theta =3\mathrm{/}2$. The allocated distortions are $(\min (4,3\mathrm{/}2),\min (1,3\mathrm{/}2))=(3\mathrm{/}2,1)$. By [F1]–[F2], $R(5\mathrm{/}2)=\frac{1}{2}({\log }_2\frac{4}{3\mathrm{/}2}+{\log }_2^{+}\frac{1}{3\mathrm{/}2})=\frac{1}{2}{\log }_2\frac{8}{3}\mathrm{.}$

Put $a=\sqrt{5\mathrm{/}2}$ and $b=\sqrt{3\mathrm{/}2}$. Choose independent standard normals $U,V,X_2$ by [F5], and set $X_1=aU+bV$, $Y_1=aU$, $Y_2=0$. The matrix with rows $(a,b)\mathrm{/}2$ and $(b,-a)\mathrm{/}2$ is orthogonal, so [F4] gives $X_1\sim N(0,4)$, independently of $X_2$. The errors are $bV$ and $X_2$, with total mean square $b^2+1=5\mathrm{/}2$.

Write $f_s(t)=(2\pi s{)}^{-1\mathrm{/}2}\exp (-t^2\mathrm{/}(2s))$ for the $N(0,s)$ density. By [F4]–[F6], $(X_1,Y_1)$ has density $f_{a^2}(y)f_{b^2}(x-y)$. Relative to Lebesgue measure in $(x_1,x_2,y_1)$ times the point mass at $y_2=0$, the joint law and product of the marginals therefore have respective positive densities $f_{a^2}(y_1)f_{b^2}(x_1-y_1)f_1(x_2)$ and $f_4(x_1)f_1(x_2)f_{a^2}(y_1)$. Their log ratio is $\frac{1}{2}\ln \frac{4}{b^2}-\frac{(x_1-y_1{)}^2}{2b^2}+\frac{x_1^2}{8}\mathrm{.}$ It is integrable since both quadratic terms have finite expectations. By [F3] and step 1.2 its joint expectation is $\frac{1}{2}\ln (8\mathrm{/}3)-\frac{1}{2}+\frac{1}{2}=\frac{1}{2}\ln (8\mathrm{/}3)$ nats. Dividing by $\ln 2$ yields the rate in step 1.1, proving attainment without assuming a Lebesgue density for the constant reconstruction coordinate.