Let $X\sim \mathcal{N}(\mu, \sigma^2I)$ where $\mu\in\mathbb{R}^n$ and $\sigma^2\in\mathbb{R}$. From subgaussian norm concentration, we know that the norm $\|X-\mu\|_2$ concentrates around $\sigma\sqrt{n}$. Is there any way to get a tail bound for the norm? Something of the form: $$\mathbb{P}(\|X - \mu\|_2 > t) \le C\exp(-cf(t, \sigma, n)).$$
I am looking for the function $f(t, \sigma, n)$. Any hints or directions to work in would help. Thanks!