Stationarity gives $H(X_n)=H(X_{n+1})=H(X_1)$. The Markov relation $X_1\to X_n\to X_{n+1}$ and the <data processing inequality for mutual information> imply
$$
I(X_1;X_{n+1})\leq I(X_1;X_n).
$$
Using $H(X_n\mid X_1)=H(X_n)-I(X_1;X_n)$ on both sides gives
$$
H(X_{n+1}\mid X_1)\geq H(X_n\mid X_1).
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 224 / 1 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-224/1/c/solution