Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 208 / 2 / b / Solution

Let $X_1,\dots ,X_N$ be independent with laws $p_1,\dots ,p_N$. We prove the claim by induction. Split the law of total variance at the last coordinate:The first term is at most $c_N^2\mathbb{E}|{\partial }_{N}f(X){|}^2$. Apply the induction hypothesis to $g(x_1,\dots ,x_{N-1})={\mathbb{E}}_{X_N}f(x_1,\dots ,x_{N-1},X_N)$. Differentiation under the expectation and Jensen inequality giveWriting $c={\max }_i{c}_i$ and combining the terms yieldsThis is the Tensorization of a Poincaré inequality.