three conjectures in statistics of iid variates Consider $N$ different i.i.d. variates $x_i$ (for $1\leq i\leq N$). Let $\hat{x}$'s be the ordered variates, in other words $\hat{x}_j$ is the $j$th largest variate. 

The density function for $\hat{x}_n$ may be computed as
$$\hat{f}_n(\alpha)=\frac{N!}{(n-1)!(N-n)!}F^{N-n}(\alpha)\Big(1-F(\alpha)\Big)^{n-1}f(\alpha)$$
in obvious notations.

What may be said about the following propositions?


The limit 
$$\lim_{N\rightarrow\infty}Nf(\langle\hat{x}_1\rangle)$$
is finite (and nonzero).
$$\text{Var}(\hat{x}_1)\geq\text{Var}(\hat{x}_2)\geq\text{Var}(\hat{x}_3)\geq\cdots\geq\text{Var}(\hat{x}_{\lfloor\frac{N}{2}\rfloor})$$
$$\lim_{N\rightarrow\infty}\hat{f}_{\lfloor rN\rfloor}=\delta(.-F^{-1}(r))\hspace{5mm}\forall r\in(0,1)$$




P.S. You may (if you want) further assume that $f$ is nonzero on $(0,+\infty)$ and is monotonously decreasing.
A counterexample to the first propositionis afforded by a continuous uniform distribution, where $f$ is constant and thus the expression under the limit is proportional to $N$.

A counterexample to the second proposition is affored by a discrete uniform distribution on three values, where for large $N$ the variance of the median goes to zero (since the median is almost certainly the middle value) whereas the variance of the tertiles remains finite in the limit (since they have finite probabilities of being the middle value or one of the outer values).

The third proposition doesn't seems implausible on the face of it, as the left-hand side is independent of $r$ and the left-hand side doesn't seem to be.
A counterexample to the first propositionis afforded by a continuous uniform distribution, where $f$ is constant and thus the expression under the limit is proportional to $N$.

A counterexample to the second proposition is affored by a discrete uniform distribution on three values, where for large $N$ the variance of the median goes to zero (since the median is almost certainly the middle value) whereas the variance of the tertiles remains finite in the limit (since they have finite probabilities of being the middle value or one of the outer values).

The third proposition doesn't seems implausible on the face of it, as the left-hand side is independent of $r$ and the left-hand side doesn't seem to be.