Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 208 / 2

Apply the Poincaré inequality in probability theory to $g=e^{\lambda f/2}$. Since $\Vert \nabla f\Vert \le 1$,HenceIterating this estimate $m$ times yieldsvalid under the stated bound on $\lambda$.

Since $\mathbb{E}f(X)=0$ and the moment-generating function is finite near zero, $\log F(u)=O(u^2)$. Consequentlywhich proves $F(\lambda /2^m{)}^{2^m}\to 1$.

For ${\lambda }_{\ast }=C_P(X{)}^{-1/2}$, part (a) and the elementary bound $-\log (1-x)\le 4x/3$ for $0\le x\le 1/4$ giveThus $F({\lambda }_{\ast })\le 3$.

The Chernoff bound and part (b) giveApply the same argument to $-f$ and use the union bound to obtain

Subtract a constant so that $f(0)=0$, and write $m=\mathbb{E}f(Y)$, $V=\mathrm{Var}f(Y)$, and $A=\mathbb{E}{f}^{\prime }(Y{)}^2$. The supplied identity applied to $f$ and $f^2$ givesBy Cauchy-Schwarz inequality, $m^2\le A$ andThus $V+m^2\le 2\sqrt{(V+m^2)A}$, so $V+m^2\le 4A$ and in particular $V\le 4A$. Therefore the standard Laplace distribution has $C_P(Y)\le 4$.