# Lower bounds on sum of squared sub-gaussians

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.