Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 208 / 3

The Gaussian logarithmic Sobolev inequality says that for standard Gaussian $X\in {\mathbb{R}}^n$ and smooth $u$,Apply it to $u=\sqrt{f}$. Since $\nabla \sqrt{f}=\nabla f/(2\sqrt{f})$,This sharp inequality immediately implies the requested weaker bound with constant $2$.

For $k\ge 0$, take $f_k(x)=1_{\{x\ge k+1\}}$ and write $q_k=\mathbb{P}(X\ge k+1)$. Since $f_k^2=f_k$,Its discrete derivative is nonzero only at $k$, soFor a Poisson distribution, $p_{k+1}/p_k=\lambda /(k+1)$ and $q_k\sim p_{k+1}$ as $k\to \infty$. Hencewhere the final estimate follows from Stirling formula. No finite constant $C$ can therefore make the proposed inequality hold for every $f$.

Let ${\epsilon }_1,\dots ,{\epsilon }_n$ be independent $\mathrm{Bernoulli}(\lambda /n)$ variables and $S_n={\sum }_i{\epsilon }_i$. Apply the tensorization of entropy to $f(S_n)$ and then apply the stated Bernoulli log-Sobolev inequality in each coordinate. If $S_n^{(i)}=S_n-{\epsilon }_i$, this givesFor each fixed $i$, $(S_n^{(i)},{\epsilon }_i)$ converges in distribution to $(X,0)$, where $X\sim \mathrm{Poisson}(\lambda )$. The Poisson limit theorem in fact gives convergence in total variation. The assumptions $K_1\le f\le K_2$ and $|Df|\le K_3$ make all displayed integrands bounded, so expectations and entropy pass to the limit. Since $n(\lambda /n)(1-\lambda /n)\to \lambda$,

Put $Y=g(X)-\mathbb{E}g(X)$ and $H(\lambda )=\log \mathbb{E}{e}^{\lambda Y}$. Apply the assumed Poisson log-Sobolev inequality to $f=e^{\lambda g}$. Since $|Dg|\le 1$ and $|e^a-1|\le \lambda e^{\lambda }$ for $|a|\le \lambda$,After division by $\mathbb{E}{e}^{\lambda g}$, the entropy bound becomesBecause $(H(\lambda )/\lambda {)}^{\prime }=(\lambda H^{\prime }-H)/{\lambda }^2$ and $H^{\prime }(0)=0$, integration from $0$ to $\lambda$ givesThe Chernoff bound and optimization over $\lambda \ge 0$ therefore yieldwhere ${\psi }^{\ast }$ is the Legendre transform of a cumulant-generating function, also called the Chernoff-Cramér transform.