Let $U_1,\dots,U_n$ be i.i.d. drawn from the uniform distribution on $[0,1]$ and let $U_{(1)} \le U_{(2)} \le \dots \le U_{(n)}$ be their order statistics. Assume that $n$ is even and that we take $U_{(1)} \le U_{(2)} \le \dots \le U_{(n/2)}$ and randomly permute them to get $\tilde U_1, \tilde U_2, \dots, \tilde U_{n/2}$. This way we get an IID sequence. Intuitively, each $\tilde U_i$ should have a distribution close to uniform $[0,1/2]$. But how far away the actual distribution is from uniform $[0,1/2]$? In particular, it seems that the support of this new distribution is still $[0,1]$, but can we control the tail of it past $1/2$?
EDIT: As has been pointed out, $(\tilde U_i)$ won't have the distribution of an IID sequence, but that of an exchangeable one. I am mainly interested in estimates for large $n$. (It would also be interesting to see how far from IID they are for large $n$, if there is a way to estimate that).