Suppose we have standard normal random variables: $X_1, X_2, ..., X_n \sim N(0,1)$ and we denote by $X_{(i)}$ the corresponding ordered statistics, i.e.
$$X_{(1)} \leq X_{(2)} \leq ... \leq X_{(n)}.$$ In particular, we have $ X_{(1)} = min(X_1, ..., X_n)$ and $ X_{(n)} = max(X_1, ..., X_n)$.
Let $\Phi$ be the cumulative distribution function of the standard normal. If I don't know $X_{(i)}$, but only its rank and distribution, intuitively the best estimate that I can do would be its quantile, $\Phi^{-1}\left( \frac{i}{n+1} \right)$, and this is the principle in QQ-plot.
Therefore, I would like to know if the following converge in probability:
$$ \frac{1}{n}\sum_{i=1}^n \left(X_{(i)} - \Phi^{-1}\left( \frac{i}{n+1} \right) \right)^2 $$
and what would be the rate of convergence.