region for a few special cases. As in the previous section, subscripts are dropped from the pmf for simplicity of notation.

Lossless Compression\[sub:RD\_BC\_lossless\]
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In this section, a single-letter characterization of the rate-cost region $\mathcal{R\mbox{\ensuremath{(0,0,\Gamma)}}}$ is derived for the special case in which the distortion metrics are assumed to be Hamming and the distortion constraints are $D_{1}=0$ and $D_{2}=0$.

\[prop:RD\_BC\_lossless\]The rate-cost region $\mathcal{R\mbox{\ensuremath{(0,0,\Gamma)}}}$ for the cascade-broadcast source coding problem illustrated in Fig. \[fig:fig3\] with Hamming distortion metrics is given by the union of all rate triples $(R_{1},R_{2},R_{b})$ that satisfy the conditions

\[eqn: RD\_BC\_lossless\] $$\begin{aligned}
R_{b} & \geq & I(X;A)\label{eq:Ra_lossless}\\
R_{1}+R_{b} & \geq & I(X;A)+H(X|A,Y)\label{eq:R1+Rb_lossless}\\
\ce{and}\mbox{ }R_{2}+R_{b} & \geq & I(X;A)+H(X|A,Z)\label{eq:R2+Rb_lossless}\end{aligned}$$

where the mutual information terms are evaluated with respect to the joint pmf $$\begin{aligned}
p(x,y,z,a)=p(x,a)p(y,z|a,x),\label{eq:joint-BC_lossless}\end{aligned}$$ for some pmf $p(a|x)$ such that $\textrm{E}[\Lambda(A)]\leq\Gamma$.

If $R_{1}=0$ and $R_{2}=0,$ the rate-cost region $\mathcal{R}(\Gamma)$ of Proposition \[prop:RD\_BC\_lossless\] reduces to the one derived in [@Weissman_multi Theorem 1].

The rate region (\[eqn: RD\_BC\_lossless\]) also describes the rate-distortion region under the more restrictive requirement of lossless reconstruction in the sense of the probabilities of error $\textrm{Pr}[X^{n}\neq\hat{X}_{j}^{n}]\leq\epsilon$ for $j=1,2$, as it follows from standard arguments (see [@Elgammal Sec. 3.6.4]). A similar conclusion applies for Corollary \[coro:cascade\_lossless\].

The converse 