Let $(X_i)_{i=1}^N$ an i.i.d. sequence (in a finite set $A$) and let $(X_{(i)})_{i=1}^n$ be a subsample of size $n \leq N$ (draw $n$ elements uniformly without replacement). I want to characterize in some sense the distribution of $(X_{(i)})_{i=1}^n$ in order to choose a $n$ such that the empirical distribution of $(X_{(i)})_{i=1}^n$ is close to the empirical distribution of $(X_i)_{i=1}^N$.
If anyone has an idea how to do this it will be of much help
Thanks!