Let Z, Z1, Z2, . . . be i.i.d. real-valued random variables. Let F be the distribution function of Z given by                                        F(t) = P{Z ≤ t} = E{I(−∞,t](Z)}, and let Fn be the...

Let Z, Z1, Z2, . . . be i.i.d. real-valued random variables. Let F be the distribution function of Z given by

                                       F(t) = P{Z ≤ t} = E{I(−∞,t](Z)},

and let Fn be the empirical distribution function of Z1, ..., Zn given by

                                 Fn(t) = #{1 ≤ i ≤ n : Zi ≤ t} n = 1 n n i=1 I(−∞,t](Zi).

(a) Show, for any 0 <>

                            P  sup t∈R |Fn(t) − F(t)| ≥  ≤ 8 · (n + 1) · exp  −n · 2 128  .

(b) Conclude from a) the Glivenko–Cantelli theorem:

                            sup t∈R |Fn(t) − F(t)| → 0 (n → ∞) a.s.

(c) Generalize (a) and (b) to multivariate (empirical) distribution functions. Hint: Apply Theorem 9.1 and Problems 9.1 and 9.4.