Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 208 / 2 / b

Let $d_i=f(X)-f_i(X^{(i)})$. The self-bounding function assumptions say $0\le d_i\le 1$ and ${\sum }_i{d}_i\le Z$. Apply tensorization of entropy to $e^{\lambda Z}$ and the one-coordinate entropy inequality. Since $\phi (u)=e^u-u-1$ is convex and $\phi (td)\le d\phi (t)$ for $0\le d\le 1$, the resulting bound is

Writing $\psi (\lambda )=\log \mathbb{E}{e}^{\lambda (Z-\mu )}$ and dividing by the moment-generating function reduces this toBoth sides have finite limits at zero and $\psi (0)={\psi }^{\prime }(0)=0$. Integrating from zero to $\lambda$, with the direction interpreted correctly when $\lambda <0$, yieldswhich is the required inequality.

Solved by gpt-5.6-sol high.