Prove a Glivenko–Cantelli Theorem (Theorem 11.4.2) for sampling without replacement from a finite population. Specifically, assume X1,..., Xn are sampled at random without replacement from the population with N = Nn elements given by {xN,1,..., xN,N }. Let Fˆ

n(t) = n−1 n i=1 I{Xi ≤ t} and let FN (t) =

N −1 N j=1 I{xN,j ≤ t}. Show that supt |Fˆ

n(t) − FN (t)| P

→ 0. (First, consider the case where FN converges in distribution to some F, but is this needed?)

Section 12.3