One useful form of the <modified logarithmic Sobolev inequality> is the following. For a function $F$ of independent coordinates, let
$$
F_i(x^{(i)})=\inf_{z\in[0,1]}F(x_1,\ldots,x_{i-1},z,x_{i+1},\ldots,x_n)
$$
and $V^+(x)=\sum_i(F(x)-F_i(x^{(i)}))^2$. If $V^+\leq v$ and $\lambda\geq0$, the inequality gives
$$
\operatorname{Ent}(e^{\lambda F})
\leq\frac{\lambda^2v}{2}\mathbb Ee^{\lambda F},
$$
and the <Herbst argument> yields
$$
\mathbb P(F-\mathbb EF\geq t)\leq e^{-t^2/(2v)}.
$$

<Talagrand's one-sided bounded differences inequality> gives the complementary tail under the same one-sided bounded-difference condition:
$$
\mathbb P(F-\mathbb EF\leq-t)\leq e^{-t^2/(2v)}.
$$
Equivalent versions use an independent coordinate replacement and its conditional positive-part variance proxy.

Solution

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

The same one-sided proxy $V^+\leq1$ lets <Talagrand's one-sided bounded differences inequality> control the opposite deviation:
$$
\mathbb P(Z-\mathbb EZ\leq-t)\leq e^{-t^2/2}.
$$
The variance estimate in part b verifies the finite-variance hypothesis in formulations of Talagrand's inequality that state it explicitly.

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

{c}

If a function of independent coordinates has one-sided squared-difference proxy at most $v$, Talagrand's one-sided inequality gives a Gaussian lower-tail bound $\mathbb P(Z-\mathbb EZ\leq-t)\leq e^{-t^2/(2v)}$. Applied to the negative of a concave function, it gives the corresponding upper-tail bound.


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 Incoming links: Talagrand's one-sided bounded differences inequality