Any hint about how does the following sum grow for k going to infinity?
$\sum_{i=1}^{k-1} \phi[\Phi^{-1}(i/k)]$
I "feel" it grows as $k/\sqrt{4\pi}$... but I am not able to prove it. I have also looked for some approximations for the inverse normal cdf.
It does look that way empirically, and would do even more if you looked at $$\lim_{k\to \infty}\dfrac{\displaystyle \sum_{i=1}^{k} \phi\left[\Phi^{-1}\left(\frac{i-\frac12}{k}\right)\right]}{k}$$ which is essentially a discrete approximation to $$E[\phi(X)] = \int_{-\infty}^\infty \phi(x)^2\, dx = \int_{-\infty}^\infty \frac{1}{2\pi}\exp\left(-{x^2}\right) dx = \frac{1}{2\sqrt{\pi}}$$ and this can be adapted to the actual question you asked.