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

One useful form of the modified logarithmic Sobolev inequality is the following. For a function $F$ of independent coordinates, letand $V^{+}(x)={\sum }_i(F(x)-F_i(x^{(i)}){)}^2$. If $V^{+}\le v$ and $\lambda \ge 0$, the inequality givesand the Herbst argument yields

Talagrand's one-sided bounded differences inequality gives the complementary tail under the same one-sided bounded-difference condition:Equivalent versions use an independent coordinate replacement and its conditional positive-part variance proxy.