tail bound of the euclidean norm of multivariate norm distributed random variable

Question

Let $x$ has a multivariate norm distribution, i.e, $x\sim \mathcal{N}(\mu,\Sigma)$, what is the following upper bound $\Pr(||x||_2^2\geq M)\leq ?$, where $M$ is a constant.

Thanks a lot!

Answer 1

$P(\|x\|_2^{2} \geq M) \leq \frac 1 M E\|x\|^{2}=\frac 1 M \sum\limits_{k=1}^{n} Ex_i^{2}=\frac 1 M \sum\limits_{k=1}^{n} (\sigma_i^{2}+\mu_i^{2})$.