Letting $\left\{X_{i}\right\}_{i=1}^{n}$ be an i.i.d. sequence of zero-mean sub-Gaussian variables with parameter $\sigma,$ define $Z_{n} :=\frac{1}{n} \sum_{i=1}^{n} X_{i}^{2} .$ Prove that $$ \mathbb{P}\left[Z_{n} \leq \mathbb{E}\left[Z_{n}\right]-\sigma^{2} \delta\right] \leq e^{-n \delta^{2} / 16} \quad \text { for all } \delta \geq 0 $$

I know some results which says that the square of sub-gaussian variable are sub-exponential and maybe apply Chernoff bounds to the sum. But I am still not be able to prove this formally. Any hints will be very helpful.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.