set of all uniquely decipherable codes on $Z_1^n$. Then, the average log-blocklength normalized redundancy of a code $\phi \in \Phi_n$ for the class of sources described by $p_{\Theta}(\theta)$ is $$\mathcal{L}_{\phi,n}(p_{\Theta}) = \int_{\mathcal{T}} \frac{r_{\phi,n}}{\log n} p_{\Theta}(\theta) \, d\theta \mbox{.}$$

The minimum $n$th-order average log-blocklength normalized redundancy is $$\mathcal{L}_{n}^{*}(p_{\Theta}) = \inf_{\psi \in \Phi_n} \mathcal{L}_{\psi,n}(p_{\Theta}) \mbox{.}$$ Finally, the minimum average log-blocklength normalized redundancy is $$\mathcal{L}^{*}(p_{\Theta}) = \lim_{n\to\infty} \mathcal{L}_{n}^{*}(p_{\Theta}).$$ If $\mathcal{L}^{*}(p_{\Theta}) = 0$, then a sequence of codes that achieves the limit is called *weighted log-blocklength normalized universal*. Now let $T$ be the set of all probability distributions defined on the alphabet $\mathcal{T}$. Then the $n$th-order maximin log-blocklength normalized redundancy of $T$ is $$\mathcal{L}_{n}^{-} = \sup_{q_{\Theta} \in T} \mathcal{L}_{n}^{*}(q_{\Theta}) \mbox{.}$$ If it exists, then the maximin log-blocklength normalized redundancy is $$\mathcal{L}^{-} = \lim_{n\to\infty}\mathcal{L}_{n}^{-} \mbox{.}$$ If $\mathcal{L}^{-} = 0$, then a sequence of codes that achieves the limit is called *maximin log-blocklength normalized universal*. The $n$th-order minimax log-blocklength normalized redundancy of $\mathcal{T}$ is $$\mathcal{L}_{n}^{+} = \inf_{\phi\in\Phi_n}\sup_{\theta\in\mathcal{T}} \frac{r_{\phi,n}(\theta)}{\log n}$$ and the minimax log-blocklength normalized redundancy of $\mathcal{T}$ is $$\mathcal{L}^{+} = \lim_{n\to\infty}\mathcal{L}_{n}^{+} \mbox{.}$$ If $\mathcal{L}^{+} = 0$, then a sequence of codes that achieves the limit is called *minimax log-blocklength normalized universal*.

### Redundancy-Capacity Theorems

The senses 