Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 208 / 2 / d / Solution

Again put $dQ=ZdP$. With ${\mathbb{E}}^{[i]}Z=\mathbb{E}[Z\mid X_i]$, the conditional density of $Q_{X^{(i)}\mid X_i}$ relative to $P_{X^{(i)}}$ is $Z/{\mathbb{E}}^{[i]}Z$. ConsequentlyPart c now gives the alternative tensorization boundUnlike the bound in part b, each summand averages over all coordinates other than $i$ while holding $X_i$ fixed.