Corollary

Let $\mathcal{P}$ denote the class of iid sources on $\mathcal{X}$. If $\mathcal{C}_{n}$ is the arithmetic code for KT distribution on $\mathcal{X}^{n}$, then its redundancy satisfies: $$\max_{p\in\mathcal{P}}R(\mathcal{C}_{n},p)\le \frac{m-1\log n}{2n}+O\left( \frac{1}{n} \right)$$This implies that KT coding distribution is minimax optimal.