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Let $X_1,\ldots,X_n$ be a sequence of i.i.d. standard Gaussian random variables. Denote the maximum of this sequence by $M_n$. I am interested in evaluating the following expectation:

$$\tag{1}\mathbb{E}\left[e^{-f(n)\sqrt{\log n}M_n}\right]$$

when $n$ is large for $f(n)$ satisfying $f(n)=\omega(1/\log\log n)$ and $f(n)=o(1)$. My best guess is that if this expectation exists, it tends to $n^{-f(n)}$ as $n\rightarrow\infty$, since I can show (see below) that this holds for $f(n)=o(1/\log\log n)$. Of course, my problem would be solved if there existed an (asymptotic) expression for the moment generating function of $M_n$, but I am not aware of such. This question seems to be related and maybe indicative on how hard this problem is. Any suggestions would be appreciated.

What I did

Using the techniques in my answer to my related question, one can show that $P\left(\lim_{n\rightarrow\infty}f(n)\log n\left(\frac{M_n}{\sqrt{2\log n}}-1\right)=0\right)=1$ when $f(n)=o(1/\log\log n)$, as $\limsup$ and $\liminf$ are both zero in that case. Arithmetic manipulation of $(1)$ then yields the asymptotic form $n^{-f(n)}$.

Another remark

An answer by Shai Covo to this question contains the reference to this paper. Using the equation for the asymptotic approximation of the $i^{\text{th}}$ order statistic found in this paper and used in the answer, we can write the following:

$$\tag{2}M_n\approx\sqrt{2\log n}-\frac{\log(\log(n))+\log(4\pi)-2W}{2\sqrt{2\log n}}$$

where $W$ is a random variable that has the density $f_W(w)=\exp[-w-\exp(-w)]$, $-\infty <w<\infty$.

Now, I am not sure whether $(2)$ can be used to obtain the asymptotic approximation to moment generating function of $M_n$. If we do, using Mathematica, we obtain the following:

$$\begin{array}{rcl}\mathbb{E}[e^{tM_n}]&\approx&\exp\left[t\left(\sqrt{2\log n}-\frac{\log(\log(n))+\log(4\pi)}{2\sqrt{2\log n}}\right)\right]\times\int_{-\infty}^\infty \exp\left[-w\left(1-\frac{t}{\sqrt{2\log n}}\right)-\exp(-w)\right]dw\\ &=&\exp\left[t\left(\sqrt{2\log n}-\frac{\log(\log(n))+\log(4\pi)}{2\sqrt{2\log n}}\right)\right]\Gamma\left[1-\frac{t}{\sqrt{2\log n}}\right] \end{array}$$ valid for $t<1$. Plugging in $t=-f(n)\sqrt{\log n}$, dropping low order terms, and absorbing the constants into $f(n)$, I get:

$$\mathbb{E}\left[e^{-f(n)\sqrt{\log n}M_n}\right]\approx n^{-f(n)}(\log n)^{f(n)}\Gamma[1+f(n)]$$

However, I am not sure if this usage of $(2)$ is kosher...

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