\tfrac{1}{2}\log\left(2\pi e \theta(1-\theta)\right) q_{\Theta}(\theta) \, d\theta \\
 &= \tfrac{1}{2}\log n + o(\log n)\mbox{,}\end{aligned}$$ where the $a_k$s are known constants given in [@JacquetS1999].

Now the mutual information is $I(\{X_i\}_{i=1}^n;\Theta) = H(\{X_i\}_{i=1}^n) - H(\{X_i\}_{i=1}^n \mid \Theta)$, so the log-blocklength normalized information rate is given by $$\begin{aligned}
\mathfrak{I}
  &= \lim_{n\to\infty} \frac{I(\{X_i\}_{i=1}^n;\Theta)}{\log n}
   = \lim_{n\to\infty} \frac{\log(n+1) - \tfrac{1}{2}\log n - o(\log n)}{\log n} \\
  &= \lim_{n\to\infty} \frac{\log(n+1) - \tfrac{1}{2}\log n}{\log n}
   = \lim_{n\to\infty} -\frac{1}{2} + \frac{\log(n+1)}{\log n}
   = \frac{1}{2}.\end{aligned}$$ Since this is greater than $0$, we have shown that the weakest form of universality is not possible, by Theorem \[thm:MIuniversal\]. Thus by Theorem \[thm:relation\], stronger forms of universality are not possible either. Since the class of binary memoryless sets is a subset of more general source classes such as memoryless; Markov; and stationary, ergodic, universal source coding over these source classes is not possible either.

We can calculate the weighted redundancy for classes of memoryless sources with larger alphabet sizes. Using the $p_{\Theta}(\theta)$ that yields equiprobable multisets and the conditional entropy given by the entropy of a multinomial random variable [@JacquetS1999], we are interested in $$\lim_{n\to\infty} \frac{\log\binom{n+|\mathcal{X}|-1}{|\mathcal{X}|-1} - \Half(|\mathcal{X}|-1)\log(K n) - o(1)}{\log n} = \frac{|\mathcal{X}|-1}{2} \mbox{,}$$ where $K$ is a known constant. As we 