(a) For i.i.d. RVs X1, . . . , Xn with CDF F find the CDF of the RV Yn := max(X1, . . . , Xn). For the remaining parts of the problem suppose that Xi , i = 1, . . . , n take values on N0 and P(X = k) = 1/2 k+1, k 0.

(b) Show that for some constant c > 1, the probability P(Yn c log2 n) 0, where Yn is defined in Part (a).

(c) Therefore prove that for some positive constants A < B (which may depend on n), the expectation of Yn satisfies A log2 n EYn B log2 n.

(d) Show that the sequence Yn/n converges to 0 in probability (please do not rely on the result of part (e)).

(e) Show that the sequence Yn/n converges to 0 a.s.