Concentration of $\ell_2$ norm of a vector sampled from a distribution

Question

Let $X=(X_1,\ldots,X_n)$, where $X_i \sim P_{p_i}(0,\frac{1}{\lambda})$ are iid, $P_{p_i}$ is sub gaussian distribution for $i^\text{th}$ element, and 0 and $1/\lambda$ are mean and variance.

I'm looking for a result on the concentration of $\|X\|_2^2$ something of the form $E \|X\|_2^2 \leq c$ with $P(\|X\|_2^2 \geq c+\epsilon)\leq f(\epsilon)$. What happens when all the distribution are normals?

I have asked a similar question on https://math.stackexchange.com/questions/3798100/concentration-of-l2-norm-of-a-vector-sampled-from-a-distribution

Answer 1

WLOG, let $\lambda = 1$ (rescale your problem appropriately, if necessary). Then, it is well-known consequence of Bernstein's inequality (e.g see theorem 3.1.1 of "High-dimensional Probability" book by R. Vershynin) that

$$ \mathbb P\left(\left|\frac{\|X\|^2}{n}-1\right| \le \epsilon\right) \ge 1 - 2e^{-Cn\min(\epsilon,\epsilon^2)},\;\forall \epsilon \ge 0. $$

Here, $C$ is a constant which is independent of $n$. In other words, $\|X\|^2$ has good (exponential / Gaussian) concentration around the value $n$.