Solve the following:

(Longest sequence of heads II. ) Let Rn := sup {k 2 0 : Xn = Xn+1 = . .. = Xnth-1 = 1}. That is, Rn denotes the length of sequnece of Is beginning at n. (If Xn = 0 then Rn = 0.) Show that Run P lim sup = |logpl-1 ) = 1. log n Hint: For every fixed parameter value of a > 0, let B@) := {Rn > a logn/|logpl}. Show that if a > 1 then with probability 1 only finitely many events Bn . If a