Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 210 / 1 / Solution

A centered random variable is sub-Gaussian with parameter ${\sigma }^2$ whenFor $\lambda >0$, the Chernoff bound givesMinimizing at $\lambda =x/{\sigma }^2$ yields $e^{-x^2/(2{\sigma }^2)}$. Applying the same argument to $-X$ proves the left-tail bound.

The moment-generating-function inequality and its version at $-\lambda$ implyComparing the second-order terms as $\lambda \to 0$ gives $\mathbb{E}{X}^2\le {\sigma }^2$. Since $\mathbb{E}X=0$,

A centered $Z$ is Sub-Gamma random variable in the right tail with variance factor $v$ and scale factor $c$ whenThe corresponding Bernstein's inequality is

A standard squared-sub-Gaussian lemma, obtained by integrating the sub-Gaussian tail or expanding exponential moments, states thatScaling a sub-Gamma variable by $a\ge 0$ multiplies its variance factor by $a^2$ and its scale by $a$; independent sums add variance factors and take the largest scale. DecomposeTheir sum is therefore sub-Gamma on the right with variance factorand scale factorApply Bernstein to the sum at threshold $nx$ to obtain