The law of large numbers says that if I sample $n$ points independently from a probability density function $f$, then the number of points lying in a neighborhood of a point $x$ with area $\epsilon$ will be proportional to $\epsilon n f(x)$. My question is: what if I sample $mn$ points from $f$, and these points are grouped into sets of size $m$? (i.e. you have $S_1 = \{x_1,\dots,x_m\}, S_2 = \{x_{n+1},\dots,x_{2m}\},\dots,S_n=\{x_{(n-1)m+1},\dots,x_{mn}\}$ There are now $m^n$ ways to select one element from each of the $n$ sets, and many of those selections may look nothing like the original distribution $f$. Is there anything we can still say probabilistically about these selections?
As a simple example, suppose that I sample $mn$ points uniformly at random on the unit interval. Suppose that for each one of the $n$ sets (consisting of $m$ points), I select the largest element. Then, the distribution of these largest points has a density function $f(x)= mx^{m-1}$. There are other ways I could select these points that would give other densities. I'd like to know if there's anything I can say about these "other density" functions in terms of $m$.