**W.I.P.: Work in progress**

Following p. 370 of Cramer's 1946 *Mathematical Methods of Statistics*, define $$\Xi_n = n(1 - \Phi(Z_n)) \,. $$ Here $\Phi$ is the cumulative distribution function of the standard normal distribution, $\mathscr{N}(0,1)$. As a consequence of its definition, we are guaranteed that $0\le \Xi_n \le n$ almost surely.

Consider a given realization $\omega \in \Omega$ of our sample space. Then in this sense $Z_n$ is both a function of $n$ and $\omega$, and $\Xi_n$ a function of $Z_n, n$, and $\omega$. For a fixed $\omega$, we can consider $Z_n$ a deterministic function of $n$, and $\Xi_n$ a deterministic function of $Z_n$ and $n$, thereby simplifying the problem. We aim to show results which hold for almost surely all $\omega \in \Omega$, allowing us to transfer our results from a non-deterministic analysis to the non-deterministic setting.

Following p. 374 of Cramer's 1946 *Mathematical Methods of Statistics,* assume for the moment (I aim to come back and supply a proof later) that we are able to show that (for any given $\omega \in \Omega$) the following asymptotic expansion holds (using integration by parts and the definition of $\Phi$):

$$ \frac{\sqrt{2\pi}}{n}\Xi_n = \frac{1}{Z_n}e^{-\frac{Z_n^2}{2}}\left( 1 + O \left( \frac{1}{Z_n^2} \right) \right) \quad ~~ as ~~ Z_n \to \infty \,.$$

Clearly we have that $Z_{n+1} \ge Z_n$ for any $n$, so $Z_n$ is almost surely an increasing function of $n$ as $n\to \infty$, therefore we claim in what follows throughout that for (almost surely all) fixed $\omega$: $$ Z_n \to \infty \quad \iff \quad n \to \infty \,. $$

Hence it follows that we have (where $\sim$ denotes [asymptotic equivalence][1]):

$$ \frac{\sqrt{2\pi}}{n} \Xi_n \sim \frac{1}{Z_n} e^{-\frac{1}{Z_n^2}} \quad ~~ as ~~ Z_n \to \infty \quad n \to \infty \,. $$

How we proceed in what follows amounts essentially to the [method of dominant balance][2], and our manipulations will be formally justified by the following lemma:

> **Lemma:** Assume that $f(n) \sim g(n)$ as $n \to \infty$, and $f(n) \to \infty$ (thus $g(n) \to \infty$). Then given any function $h$ which is formed via compositions, additions, and multiplications of logarithms and power laws (essentially any "[polylog][3]" function), we must have also that as $n \to \infty$: $$ h(f(n)) \sim h(g(n)) \,. $$ In other words, such "polylog" functions [preserve asymptotic equivalence][4].

The truth of this lemma is a consequence of [Theorem 2.1. as written here][5].

Taking logarithms of both sides, we get that:

$$\log ( \sqrt{2\pi} \Xi_n ) - \log n \sim -\log Z_n - \frac{Z_n^2}{2} \,. \tag{1}$$

This is where Cramer is somewhat cagey; he just says "assuming $\Xi_n$ is bounded", we can conclude blah blah blah. But showing that $\Xi_n$ is suitably bounded almost surely seems to be actually somewhat non-trivial. It seems that the proof of this may essentially be part of what's discussed on pp. 265-267 of Galambos, but I am not sure given that I am still working to understand the content of that book.

Anyway, **assuming one can show that $\log \Xi_n = o(\log n)$, presumably via $\Xi_n = o(n)$ almost surely**, then it follows (since the $-Z_n^2/2$ term dominates the $-\log Z_n$ term) that:

$$ - \log n \sim - \frac{Z_n^2}{2} \quad \implies \quad Z_n  \sim \sqrt{2 \log n} \,. $$

This is somewhat nice, since it is already most of what we want to show, although again it is worthwhile to note that it is essentially only kicking the can down the road, since now we have to show some certain almost surely boundedness of $\Xi_n$. On the other hand, $\Xi_n$ has the same distribution for any maximum of i.i.d. continuous random variables, so this may be tractable.

Anyway, if $Z_n \sim \sqrt{2 \log n}$ a.s., then clearly one can also conclude that $Z_n \sim \sqrt{2 \log n}(1 + \alpha(n))$ for any $\alpha(n)$ which is $o(1)$ as $n \to \infty$. Using our lemma about polylog functions preserving asymptotic equivalence above, we can substitute this expression back into $(1)$ to get:

$$\log(\sqrt{2 \pi} \Xi _n)- \log n \sim -\log (1 + \alpha) - \frac{1}{2}\log 2 - \frac{1}{2}\log \log n - \log n - 2 \alpha \log n  - \alpha^2 \log n \,.  $$

$$ \implies -\log(\Xi_n \sqrt{2 \pi}) \sim \log(1 + \alpha) + \frac{1}{2} \log 2 + \frac{1}{2} \log \log n + 2\alpha \log n + \alpha^2 \log n \,. $$ 

Here we have to go even further, and assume that $\log \Xi_n = o( \log \log n) ~~ as ~~ n \to \infty$, presumably via $\Xi_n = o(\log n)$, almost surely. Again, all Cramer says is "assuming $\Xi_n$ is bounded". But since all one can say a priori about $\Xi_n$ is that $0 \le Xi_n \le n$ a.s., it hardly seems clear that one should have $\Xi_n = O(1)$ almost surely, which seems to be the substance of Cramer's claim.

But anyway, assuming one believes that, then it follows that the dominant term which does not contain $\alpha$ is $\frac{1}{2} \log \log n$. Since $\alpha = o(1)$, it follows that $\alpha^2 = o(\alpha)$, and clearly $\log ( 1 + \alpha) = o (\alpha) = o(o(\alpha \log n))$, so the dominant term containing $\alpha$ is $2 \alpha \log n$. Therefore, we can rearrange and (dividing everything by $\frac{1}{2}\log\log n$ or $2 \alpha \log n$) find that 

$$ - \frac{1}{2} \log \log n \sim 2 \alpha \log n \quad \implies \quad \alpha \sim - \frac{\log \log n}{4 \log n} \,. $$

Therefore, substituting this back into the above, we get that:

$$Z_n \sim \sqrt{2 \log n}- \frac{\log\log n}{2 \sqrt{2 \log n}} \,, $$

again, assuming we believe certain things about $\Xi_n$.


  [1]: https://artofproblemsolving.com/wiki/index.php?title=Asymptotic_equivalence
  [2]: http://leto.net/math/latex/perturb.pdf
  [3]: https://en.wikipedia.org/wiki/Polylogarithmic_function
  [4]: https://math.stackexchange.com/questions/1671090/functions-that-preserve-asymptotic-equivalence
  [5]: http://www.math.uconn.edu/~kconrad/blurbs/analysis/asymp.pdf