Order statistics for discrete uniform random variables Let $X_i, i=1,\cdots,N$ be i.i.d. discrete uniform random variables, taking values in the range $\{0,1,...,M-1\}$. 
Let $X_{(i)}$ denote the $i$-th order statistic. 
What are the values of $\displaystyle\mathbb{E}\left[ \sum_{i=1}^{N/2} X_{(i)}\right]$ and $\displaystyle \mathbb{E}\left[\sum_{i=N/2 +1}^N X_{(i)}\right]$ when $N$ is large? 

I did some Monte Carlo simulations. 
It seems that  $$\displaystyle\frac{\displaystyle \mathbb{E}\left[ \sum_{i=1}^{N/2} X_{(i)}\right]}{\displaystyle \mathbb{E}\left[\sum_{i=N/2 +1}^N X_{(i)}\right]}$$ converges to some value, but I am not able to obtain any analytical expression.
 A: For fixed $M$, for large $N$ most of the $X_{(i)}$ have an almost certain value; only a fraction of order $N^{\frac12}$ near the values $i=k\frac NM$ with integer $k$ have an appreciable probability to take one of two different values. These only yield a contribution of order $N^{-\frac12}$ to the ratio, so we can disregard them in the limit $N\to\infty$ and just calculate with $\frac NM$ instances of each value. Then for even $M$
\begin{align}
\frac{E\left[\sum_{i=1}^{N/2}X_{(i)}\right]}{E\left[\sum_{i=N/2+1}^NX_{(i)}\right]}
\to_{N\to\infty}{}&
\frac{\sum_{m=0}^{M/2-1}m}{\sum_{m=M/2}^{M-1}m}
\\
={}&
\frac{\frac12\frac M2\left(\frac M2-1\right)}{\frac12M(M-1)-\frac12\frac M2\left(\frac M2-1\right)}
\\
={}&
\frac{M(M-2)}{4M(M-1)-M(M-2)}
\\
={}&
\frac{M-2}{3M-2}
\\
={}&
\frac13-\frac4{3(3M-2)}\;,
\end{align}
and for odd $M$
\begin{align}
\frac{E\left[\sum_{i=1}^{N/2}X_{(i)}\right]}{E\left[\sum_{i=N/2+1}^NX_{(i)}\right]}
\to_{N\to\infty}{}&
\frac{\sum_{m=0}^{(M-3)/2}m+\frac12\frac{M-1}2}{\frac12\frac{M-1}2+\sum_{m=(M+1)/2}^{M-1}m}
\\
={}&
\frac{\frac12\frac{M-1}2\frac{M-3}2+\frac12\frac{M-1}2}{\frac12\frac{M-1}2+\frac12M(M-1)-\frac12\frac{M+1}2\cdot\frac{M-1}2}
\\
={}&
\frac{(M-1)(M-3+2)}{(M-1)(2+4M-(M+1))}
\\
={}&
\frac{M-1}{3M+1}
\\
={}&
\frac13-\frac4{3(3M+1)}\;.
\end{align}
