Stationary block rate-distortion limit and endpoint continuity

Write $a_n(D)=n{R}_n(D)$, and choose minimizing channels for blocks of lengths $m$ and $n$, available by [F5]. Use their conditional product as a channel on the concatenated block. Stationarity gives the same marginal laws for the second block as for a block of length $n$; each part has mean distortion at most $D$, so the concatenated normalized distortion is at most $D$.

A best constant reconstruction has normalized expected distortion $D_{\mathrm{c}}$ and zero mutual information for every block. Applying the finite log-sum inequality [F4] to the joint pmf and the product of its marginals, both of total mass one, gives nonnegative mutual information; hence $R_n(D)=0$ for $D\ge D_{\mathrm{c}}$. If $D<D_{\min }$, every coordinate distortion is at least its pointwise minimum, so no block channel is feasible and each $R_n(D)=+\mathrm{\infty }$.

Suppose $D_{\mathrm{c}}>D_{\min }$, and let $\gamma$ be the smallest positive value among the finite gaps $d(x,\widehat{x})-{\min }_{z}d(x,z)$; thus $\gamma >0$. Fix $0<ϵ<\gamma \mathrm{/}2$. Since $D_{\min }+ϵ$ is feasible, [F5] supplies a minimizing length-$n$ channel at that distortion budget. Let $S_i=1$ when its chosen reproduction has positive gap from the pointwise minimum, and let $\alpha =n^{-1}{\sum }_i\mathbb{P}(S_i=1)$. The excess above the block minimum is at least $\gamma {\sum }_i{S}_i\mathrm{/}n$, so $\alpha \le ϵ\mathrm{/}\gamma$.

Let $(X_1,Y_1)$ and $(X_2,Y_2)$ denote the two source/output block pairs under that channel. Conditional independence of the two channel outputs given the full source block gives $H(Y_1,Y_2\mid X_1,X_2)=H(Y_1\mid X_1)+H(Y_2\mid X_2)$. Entropy subadditivity gives $H(Y_1,Y_2)\le H(Y_1)+H(Y_2)$. Subtracting these finite quantities and using [F3] yields $I(X_1,X_2;Y_1,Y_2)\le I(X_1;Y_1)+I(X_2;Y_2)$, so $a_{m+n}(D)\le a_m(D)+a_n(D)$. The source blocks need not be independent.

Fix a pointwise minimizer ${\widehat{x}}_{\ast }(x)$ for each source letter and a dummy symbol ${\widehat{x}}_{\mathrm{d}}\in \widehat{\mathcal{X}}$. Keep the original output where $S_i=0$ and replace it by ${\widehat{x}}_{\ast }(X_i)$ where $S_i=1$, obtaining ${\widehat{X}}^{\mathrm{\prime }n}$; then $d(X_i,{\widehat{X}}_i^{\mathrm{\prime }})={\min }_{\widehat{x}}d(X_i,\widehat{x})$ for every $i$, so stationarity gives $\mathbb{E}{d}_n(X^n,{\widehat{X}}^{\mathrm{\prime }n})=D_{\min }$. Let $C_i={\widehat{x}}_{\ast }(X_i)$ when $S_i=1$ and ${\widehat{x}}_{\mathrm{d}}$ otherwise. Since ${\widehat{X}}^{\mathrm{\prime }n}$ is a function of $({\widehat{X}}^n,S^n,C^n)$, data processing and the entropy chain rule give $I(X^n;{\widehat{X}}^{\mathrm{\prime }n})\le I(X^n;{\widehat{X}}^n,S^n,C^n)\le I(X^n;{\widehat{X}}^n)+H(S^n)+H(C^n\mid S^n)\mathrm{.}$ Each $C_i$ is fixed when $S_i=0$ and has at most $\mathrm{\mid }\widehat{\mathcal{X}}\mathrm{\mid }$ values when $S_i=1$, so $H(C^n\mid S^n)\le {\sum }_i\mathbb{P}(S_i=1){\log }_2\mathrm{\mid }\widehat{\mathcal{X}}\mathrm{\mid }$. Entropy subadditivity and concavity of binary entropy give $H(S^n)\le n{h}_2(\alpha )$. Thus $R_n(D_{\min })\le R_n(D_{\min }+ϵ)+h_2(ϵ\mathrm{/}\gamma )+(ϵ\mathrm{/}\gamma ){\log }_2\mathrm{\mid }\widehat{\mathcal{X}}\mathrm{\mid }$, since $h_2$ is increasing on $[0,1\mathrm{/}2]$.

Fix $k\ge 1$, write $n=qk+r$ with $0\le r<k$, and put $a_0=0$. Repeated subadditivity gives $a_n\le q{a}_k+a_r$. The finite set $\{a_r:0\le r<k\}$ is bounded, so ${\mathrm{lim\hspace{0.17em}sup}}_n{a}_n\mathrm{/}n\le a_k\mathrm{/}k$. Taking the infimum over $k$, while ${\inf }_k{a}_k\mathrm{/}k\le {\mathrm{lim\hspace{0.17em}inf}}_n{a}_n\mathrm{/}n$, proves $a_n\mathrm{/}n\to {\inf }_k{a}_k\mathrm{/}k$.

For each fixed $n$, the feasible block-channel sets are nested as $D$ increases, so $R_n(D)$ is nonincreasing. Mixing two block channels and applying [F4] shows $R_n$ is convex. Passing to the pointwise limit from step 3.1 preserves both monotonicity and the convexity inequality.

Taking infima over $n$ in step 2.2 gives the same inequality for $R_{\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k}}$. The two error terms tend to zero as $ϵ\downarrow 0$; monotonicity from step 4.1 gives the reverse bound $R_{\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k}}(D_{\min }+ϵ)\le R_{\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k}}(D_{\min })$, proving right-continuity. If $D_{\mathrm{c}}=D_{\min }$, step 1.2 already gives zero on the entire feasible interval.