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

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

• I think the first sentence should read “where the $X_i$ are iid subgaussian variables with mean 0 and finite variance.” If they are all iid, then the notation $P_{p_i}$ is needlessly complicated, and in any case the use of $1/\lambda$ seems needlessly complicated too.
– user44143
Commented Aug 24, 2020 at 6:12
• I agree with Matt F. that your notation is unnecessarily clumsy. In any case, are you aware that $\|X\|^2$ concentrates around $n\mbox{var}(X_1)$ ? Commented Aug 24, 2020 at 8:17
• @MattF. Thank you for your suggestions, I will change it. Commented Aug 24, 2020 at 17:49
• @dohmatob I don't know about that concentration. Commented Aug 24, 2020 at 17:49
• Isn't this exactly the same question your asked here math.stackexchange.com/questions/3798100/… ? Commented Aug 24, 2020 at 17:58

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 $$n$$.