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Suppose that $X_1, \ldots, X_n \sim N(0,1)$ are independent random variables. I am interested in finding a constant C that satisfies:

$$ E\left[\max_{1\leq i\leq n}|X_i|\right] \leq C \sqrt{log\ n} $$

I know one method is to employ the moment generating function trick, then take logs of both sides. However, I was wondering if there exists a more direct method. thanks!

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  • $\begingroup$ Relevant: math.stackexchange.com/questions/987604/… (gives more than the inequality) $\endgroup$ – Clement C. Sep 19 '16 at 21:42
  • $\begingroup$ @ClementC. It appears that the question doesn't involve the absolute value of $X_i$. Am I dealing with a folded normal distribution here? thanks! $\endgroup$ – user321627 Sep 19 '16 at 23:00
  • $\begingroup$ The upper bound $$P(|X_1|>x)\leqslant\frac1{x\sqrt{2\pi}}e^{-x^2/2}$$ should suffice, together with $$E(\max|X_i|)=\int_0^\infty P(\max|X_i|>x)dx$$ and $$P(\max|X_i|<x)=P(|X_1|<x)^n$$ $\endgroup$ – Did Sep 20 '16 at 6:56

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