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So I've been currently studying this book on high dimensional probability by Roman Vershynin, which I find pretty awesome! However, I was wondering if in the first line of the proof of Theorem 3.1.1 (P. 43), where he proved a concentration inequality for high dimensional random vectors with independent components, it should be "it follows that" instead of "for simplicity, we can assume that"? This is because, since he assumed that $EX_i^2=1 \forall i,$:

$$2 \ge E[e^\frac{{X_i^2}}{||X_i||_{\psi_2}^2}] \ge E[1 + \frac{{X_i^2}}{||X_i||_{\psi_2}^2}] = 1 + \frac{{EX_i^2}}{||X_i||_{\psi_2}^2} = 1 + \frac{1}{||X_i||_{\psi_2}^2},$$ implying ${||X_i||_{\psi_2}^2} \ge 1 \forall i$.

Just checking to make sure, thanks!

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    $\begingroup$ Yes, your argument is correct. $\endgroup$
    – user58955
    Mar 22, 2020 at 2:28
  • $\begingroup$ @user58955 Thanks for checking! $\endgroup$ Mar 22, 2020 at 2:34
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    $\begingroup$ You can get proper norm bars by using \| instead of ||. You can get properly sized brackets (and other paired delimiters) that adapt to the size of their content by preceding them with \left and \right. $\endgroup$
    – joriki
    Mar 22, 2020 at 4:40
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    $\begingroup$ @joriki will do that from now on - thanks for pointing out! $\endgroup$ Mar 22, 2020 at 11:40

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