Massart and Laurent (see [1], Lemma 1 on page 1325] give tail bounds for $\chi^2$ random variables.
A corollary of their bound is the following:
$$P\left[\frac{1}{k} X \leq 1- 2\sqrt{\frac{x}{k}} \right]\leq \exp (-x) $$ $$P\left[\frac{1}{k} X \geq 1+2\sqrt{\frac{x}{k}}+2 \frac{x}{k} \right]\leq \exp (-x) $$
I am looking for similar statements for sub-Gaussian and sub-exponential random variables, but could only find the following statement for independent $\sigma^2$-sub-Gaussian RV:
$$P\left[ \frac{1}{k} \sum_{i=1}^{n} X_i \geq t \right]\leq \exp \left(-\frac{n t^2}{\frac{2}{n}\sum_{i=1}^{n} \sigma_i^2}\right) $$
which is similar to the second concentration inequality ("upper tail") by Massart and Laurent. I couldn't find any equivalent for the first one ("lower tail").
Furthermore, I couldn't find any inequalities for sub-exponential RV without any additional prerequisites (e.g., maximum known).
Is there any similar statements to the first ("upper") bound by Massart and Laurent for sub-Gaussian distributions and similar statements for both bounds for sub-exponential distributions?
Thank you for your help!