For $Q_t=(1-t)Q^*+tQ^0$, convexity gives $Q_t\in E$. Minimality at $t=0$ implies
$$
0\leq\left.\frac d{dt}D(P\Vert Q_t)\right|_{0+}
=\sum_aP(a)\left(1-\frac{Q^0(a)}{Q^*(a)}\right).
$$
Since $P$ has full support and the minimum is finite, $Q^*$ has full support.


Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 224 / 3 / b / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-224/3/b/i/solution