While working on a research problem I got stuck at the following question. Suppose I have a mean zero multivariate Gaussian random vector $\mathbf{X}\sim N\left(\mathbf{0}_{n\times 1}, \Sigma_{n\times n}\right)$, with subgaussian parameter $\sigma^2$. By definition, for any $\boldsymbol{t}\in\mathbb{R}^n$, \begin{equation*} \mathbb{E}\left[\exp\left(\langle \boldsymbol{t},\mathbf{X}\rangle\right)\right]\leq \exp\left(\frac{\sigma^2||\boldsymbol{t}||_2^2}{2}\right). \end{equation*} Does the squared norm of $\mathbf{X}$ satisfy a lower tail concentration bound? That is, is there some result of the following form: \begin{equation*} \mathbb{P}\left(||\mathbf{X}||_2^2 \leq \mathbb{E}[||\mathbf{X}||_2^2] - t\right) \leq f(t,\sigma^2,n), \end{equation*} for some concrete function $f$ and any positive real number $t$? I can show that an upper tail concentration bound exists that looks like: \begin{equation*} \mathbb{P}\left(||\mathbf{X}||_2 \geq t\right) \leq 2\exp\left(\frac{-t^2}{\sigma^2 n}\right). \end{equation*} (I may be missing some constant in the numerator of the argument in the $\exp(\cdot)$ function, but it does not matter for my application. See, e.g., Lemma 1 of this note or Exercise 6.3.5 of Vershynin's High-Dimensional Probability [link]).
My ultimate goal is to work out a concentration result of the form \begin{equation*} \mathbb{P}\left(\left|||\mathbf{X}||_2^2 - \mathbb{E}[||\mathbf{X}||_2^2]\right| \geq t\right) \leqslant g(t,\sigma^2,n). \end{equation*}
And to clarify: I have some intuition from concentration of chi-square and sub-exponential random variables (e.g., tail decays like $\exp(-t)$ --- see Proposition 2.7.1 of Vershynin), so I expect it to be similar here. Except that I cannot derive the lower tail concentration bound.
Thanks!
