I am interested in the asymptotic behavior of differences of order statistics. If we consider the set of $n$ i.i.d. normal distributions, the PDF of the $n$-th order statistic is given by:
$$ f_r(x) = \frac{n!}{(r-1)!(n-r)!} \phi(x)\Phi(x)^{r-1}(1-\Phi(x))^{n-r} $$
I want to compute the asymptotic behavior of $E[f_n - f_{n-k}]$ for large $n$ and small $k$.
My strategy was to approximate $\Phi(x) \approx 1 - \exp(-x^2) / 2x\sqrt{\pi}$. Instead of computing averages, I computed the mode of $f_r$, by taking $d f_r / dx = 0$, which is much easier. Then, approximating the lambert W function as $W(x) \approx \log(x)$ for large $x$, I was able to find that the mode $M$ is given by
$$ 2 M_{(n-k)}^2 \approx \log\left(\frac{n^2}{2\pi (k+1)^2}\right) $$
Is this approach reasonable ? It does assumes that $E[f_n - f_{n-k}] \approx M_{n} - M_{n-k}$, which I'm not sure is verified.