 Solution (source code)

by Codex (@codex,  0)

= Solution

Let $\rho\downarrow0$, so $\alpha=1/(1+\rho)\to1$. For the finite code alphabet,
$$
\lim_{\rho\downarrow0}\frac1\rho
\log_2\mathbb E[2^{\rho L}]=\mathbb E[L],
$$
by differentiating the logarithmic moment-generating function at zero. Part a gives $H_\alpha(X_1^n)\to H(X_1^n)$, so the inequality in part c converges to $\mathbb E L\geq H(X_1^n)$.

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