Regularity conditions for the approximation of the mean of order statistics (i.e. when is $E[X_{(pn)}] \approx F^{-1}(p) $) Let $X_{(1)} \le X_{(2)}\le ... \le X_{(n)}$ be a sequence of order statistics generate from an i.i.d. sequence $X_1, X_2, ... , X_{n}$. We are interested in expected value of $X_{(pn)}$, for $p \in (0,1)$, that is
\begin{align}
E[X_{(pn)}],
\end{align}
we assume $pn$ is always an integer or just consider $E[X_{( \lceil pn \rceil)}]$.
A commonly used, large $n$, approximation of expectation  is
\begin{align}
E[X_{(pn)}] \approx F^{-1}(p) 
\end{align}
where $F$ is cdf of the original distribution and $F^{-1}$ the quantile function (inverse cdf).
Question: Under what regularity conditions  is this approximation valid?  In other words, under what regularity conditions do we have that
\begin{align}
\lim_{n \to \infty} | E[X_{(pn)}] - F^{-1}(p) |=0?
\end{align}
What Have I done: I have looked at books like "First Course on Order Statistics" by Bary Arnold and other references. While these approximations appear there and in other sources, I was not able to find conditions under which this approximation is valid.
The proof that I was able to find  goes as follows:
\begin{align}
E[X_{(pn)}]=&E[F^{-1}(U_{(pn)})] \text{ where  $U_{(pn)}$ is order statistics of uniform distribution}\\
&=E[F^{-1}(p) -\frac{d}{du}F^{-1}(u)|_{u=D} (p-U_{(pn)})]
\end{align}
where in the above we used Taylor's reminder theorm and where the random variable $D$ is between $p$ and $U_{(pn)}$.  This leads to
\begin{align}
E[X_{(pn)}]
&=F^{-1}(p) - E \left[\frac{1}{f(F^{-1}(D))} (p-U_{(pn)})\right]
\end{align}
where we used that $\frac{d}{du}F^{-1}(u)=\frac{1}{f(F^{-1}(u))}$ and where $f$ is pdf of $F$.
At this point we need to show that
\begin{align}
\lim_{n \to \infty} E \left[\frac{1}{f(F^{-1}(D))} (p-U_{(pn)})\right]=0.
\end{align}
However, I was not able to find conditions that guarantee this.  The key is to flip the expectation and limit, but I was not able to find a dominating random variable for this.

*

*From the consistency of estimator of quantile(c.f. A.W. van der Vaart,  Asymptotic Statistics, Cambridge University Press 1998. p.43, p.304, \$21.1),
if $F$ is strictly increasing and continuous in a neighbourhood $(F^{-1}(p)-\delta, F^{-1}(p)+\delta)$,  then
\begin{equation*}
 X_{n,([pn])}\overset{pr}{\longrightarrow}F^{-1}(p),\qquad \text{as }n\to\infty.\tag{1}
\end{equation*}


*If (1) is true and $\{  X_{n,([pn])},n\ge 1 \}$ is uniformly integrable,
then
\begin{equation*}
 \lim_{n\to\infty}\mathsf{E}[X_{n,([pn])}]=F^{-1}(p). \tag{2}
\end{equation*}


*Suppose that (i) $F$ is strictly increasing and continuous in a neighbourhood $(F^{-1}(p)-\delta, F^{-1}(p)+\delta)$,
(ii)
\begin{equation*}
 -\infty<a=\inf\{x:F(x)>0\}\le \sup\{x:F(x)<1\}=b<\infty, \tag{3}
\end{equation*}
then (2)  is true.
Proof: From (3), $a\le X_{n,([pn])}\le b $.
