Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 224 / 2 / d / Solution

Let $\rho \downarrow 0$, so $\alpha =1/(1+\rho )\to 1$. For the finite code alphabet,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 $\mathbb{E}L\ge H(X_1^n)$.