Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 208 / 4 / c

For convex $f$, the subgradient inequality givesTaking the positive supremum over $z\in [0,1]$ and summing squares showsThe modified logarithmic Sobolev inequality from part a with $v=1$ therefore gives

The same one-sided proxy $V^{+}\le 1$ lets Talagrand's one-sided bounded differences inequality control the opposite deviation:The variance estimate in part b verifies the finite-variance hypothesis in formulations of Talagrand's inequality that state it explicitly.

The function $-g$ is convex and $1$-Lipschitz. Applying part ii to $-g$ turns its lower-tail event into the upper-tail event for $g$:This proves probability (iii).

Applying part i to the convex function $-g$ givesThis proves probability (iv). Thus both requested tails of the concave function have the claimed bound.