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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

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  • $\begingroup$ 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. $\endgroup$
    – user44143
    Commented Aug 24, 2020 at 6:12
  • $\begingroup$ I agree with Matt F. that your notation is unnecessarily clumsy. In any case, are you aware that $\|X\|^2$ concentrates around $n\mbox{var}(X_1)$ ? $\endgroup$
    – dohmatob
    Commented Aug 24, 2020 at 8:17
  • $\begingroup$ @MattF. Thank you for your suggestions, I will change it. $\endgroup$
    – newbie
    Commented Aug 24, 2020 at 17:49
  • $\begingroup$ @dohmatob I don't know about that concentration. $\endgroup$
    – newbie
    Commented Aug 24, 2020 at 17:49
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    $\begingroup$ Isn't this exactly the same question your asked here math.stackexchange.com/questions/3798100/… ? $\endgroup$
    – dohmatob
    Commented Aug 24, 2020 at 17:58

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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 $n$.

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