For $\lambda\in\mathbb R$, convexity of $u\mapsto e^{\lambda u}$ on $[0,1]$ gives
$$
e^{\lambda u}\leq1-u+ue^\lambda.
$$
Consequently
$$
\mathbb Ee^{\lambda X_i}\leq1-\mu_i+\mu_i e^\lambda.
$$
For $\lambda\geq0$, independence and concavity of $u\mapsto\log\{1+u(e^\lambda-1)\}$ yield
$$
\mathbb Ee^{\lambda n\bar X}
\leq\prod_{i=1}^n\{1-\mu_i+\mu_i e^\lambda\}
\leq\{1-\bar\mu+\bar\mu e^\lambda\}^n.
$$
The <Chernoff bound> therefore gives, for $a=\bar\mu+x$,
$$
\mathbb P(\bar X\geq a)
\leq\inf_{\lambda\geq0}
\exp\left[n\{\log(1-\bar\mu+\bar\mu e^\lambda)-\lambda a\}\right].
$$
For $\bar\mu<a<1$, the minimizer satisfies
$$
e^{\lambda_*}=\frac{a(1-\bar\mu)}{\bar\mu(1-a)}.
$$
Substitution gives the <binary relative entropy>
$$
\mathbb P(\bar X-\bar\mu\geq x)
\leq e^{-n\operatorname{kl}(\bar\mu+x,\bar\mu)}.
$$
The endpoint cases follow by continuity.

Put $a=(1+\delta)\bar\mu$. The same calculation, followed by $\log(1-z)\leq-z$, gives
$$
\operatorname{kl}((1+\delta)\bar\mu,\bar\mu)
\geq\bar\mu\{(1+\delta)\log(1+\delta)-\delta\}.
$$
Thus
$$
\mathbb P\{\bar X\geq(1+\delta)\bar\mu\}
\leq
\left\{\frac{e^\delta}{(1+\delta)^{1+\delta}}\right\}^{n\bar\mu}.
$$
The supplied lower bound $\log(1+\delta)\geq2\delta/(2+\delta)$ implies
$$
(1+\delta)\log(1+\delta)-\delta
\geq\frac{\delta^2}{2+\delta},
$$
which proves the second upper-tail estimate. If $(1+\delta)\bar\mu>1$, the event is empty and the same bound remains true.

For the lower tail, apply the exponential-moment argument with $\lambda<0$, or equivalently optimize at $a=(1-\delta)\bar\mu$. It gives
$$
\mathbb P\{\bar X\leq(1-\delta)\bar\mu\}
\leq
\left\{\frac{e^{-\delta}}{(1-\delta)^{1-\delta}}\right\}^{n\bar\mu}.
$$
Finally,
$$
\delta+(1-\delta)\log(1-\delta)\geq\frac{\delta^2}{2}
$$
for $0\leq\delta<1$, yielding $e^{-n\delta^2\bar\mu/2}$; the case $\delta=1$ follows by a limit.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 210 / 1 / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-210/1/solution