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.
From Proposition 2.14, we can obtain that $$\mathbb{P}[Z_n \le \mathbb{E}[Z_n] - \sigma^2 \delta] \le exp\left(-\frac{n \sigma^2 \delta^2}{\frac{2}{n}\sum_{i=1}^n \mathbb{E}[X_i^4]}\right).$$
Then to figure out the final tail bound, we need to show that $$\mathbb{E}[X_1^4] \le 8 \sigma^4.$$
The last claim is from the sub-Gaussian assumption, w.l.o.g. assume that $\sigma^2 = 1$,
$$\mathbb{E}X^4 = \int_0^\infty 4 y^3 \mathbb{P}[|X|>y]{\rm{d}}y \le \int_0^\infty 8y^3 e^{-y^2/2}{\rm{d}}y = \int_0^\infty 4te^{-t/2}{\rm{d}}t =16,$$
which may not be the optimal constant.