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Let $X=\max\{X_1, X_2, \cdots, X_N\}$, where each $X_i \sim N(0,1)$ and are independent. What is the approximate value of $X$ for large $N$.

The term "approximate" isn't defined very clearly. I'm assuming it means the mean, but I can't really figure out a nice way to calculate that, since the PDF of the $N$th order statistic uses CDF of the normal.

Any thoughts would be appreciated.

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The distribution (asymptotic and exact) of the maximum order statistic from an iid normal sample is discussed here: Determine the limiting distribution of Standard Normal order statistics. Then the expectation can be found by numerical integration (example given there), or the asymptotic solution given there, which is a Gumbel distribution, can be used. That wikipedia page gives the expectation of a standard Gumbel variate as the Euler-Mascheroni constant $\gamma\approx 0.57721$, so with notation from the first linked page, the approximate expectation of the maximum of $n$ iid $\mathcal{N}(0,1)$ variables is $$ a_{(n)} + b_{(n)} \gamma$$.

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This question was posed in a different form some years ago on math Stack Exchange, where it was asked to prove that the maximum of $n$ independent standard Normals was asymptotically equivalent to $\sqrt{2 \log n}$.

After deriving the exact distribution of the maximum order statistic, it became immediately apparent that for large $n$ (and by that I do not mean just $n = 100$, but say $n = 1,000,000$ or even a billion), the proposed asymptote was appallingly useless, and moreover that there is no meaningful approximate value for $X$. The value of $X$ does not converge to any constant for large $n$.

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The full posting on math SE is at:

https://math.stackexchange.com/questions/961780/prove-that-the-maximum-of-n-independent-standard-normal-random-variables-is-a

In brief, in answer to your question: there is no meaningful approximate value of $X$ for large $n$.

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I think you want to look at the Fisher-Tippet Theorem which provides a CLT for the maximum, which states that the normalised maximum converges almost surely to one of three distributions (Gumbel, Frechet or Weibull)

sources: https://en.wikipedia.org/wiki/Extreme_value_theory http://www.maths.manchester.ac.uk/~saralees/chap1.pdf

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    $\begingroup$ A key aspect of the Fisher-Tippet-Gnedenko Theorem is that the maximum has to be suitably normalized for the limit to exist. In particular, as $N\to\infty$, the maximum of $N$ iid Normal distributions diverges: it does not approach any particular distribution. $\endgroup$
    – whuber ♦
    Commented Jan 23, 2018 at 18:31

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