that achieves $\mathfrak{H}(\mathfrak{Z})$.

### Log-Blocklength Normalized Redundancy Measures

Define the redundancy of a source code, $r_{\phi,n}$, as the excess average codeword length that is required over the minimum $H(Z_1^n)$: $$r_{\phi,n} = C_{\phi,n} - H(Z_1^n) \mbox{.}$$ Finally, define the log-blocklength normalized redundancy of a sequence of source codes as $$\lim_{n\to\infty}\frac{r_{\phi,n}}{\log n} = \lim_{n\to\infty}\frac{C_{\phi,n} - H(Z_1^n)}{\log n} 
= \lim_{n\to\infty}\frac{C_{\phi,n}}{\log n} - \mathfrak{H}(\mathfrak{Z}) \stackrel{\Delta}{=} \mathfrak{C}_{\phi} - \mathfrak{H}(\mathfrak{Z}) \mbox{.}$$ By the manipulations of the source coding theorem that we had made previously, we know that there is a sequence of codes with $\mathfrak{C} = 0$. The code used to develop the upper bound in the source coding theorem, however, requires that $p_{Z_1^n}(z_1^n)$ is known.

Now we define performance measures for source coding for a class of source distributions, rather than just a single source distribution. The definitions parallel those of [@Davisson1973]. Suppose that the source distribution is chosen from a class that is parameterized by $\Theta \in \mathcal{T}$. For each $\theta$, there is a conditional distribution $$p(z_1^n \mid \theta) = \Pr\left[Z_1^n = z_1^n \mid \Theta = \theta\right] \mbox{.}$$ The parameter $\theta$ is fixed but unknown, when generating the source realization. Moreover, there may be a distribution on this parameter, $p_{\Theta}(\theta)$. Let $\Phi_n$ be the 