1.9. Let {Y k' 1 S k S n} be independent r. v. 's with mean zero and finite

. 2 varIances CJ k;

k k Sk = LYj, S~ = LCJ;>O, Zk = S~ -s~

j=l j=l Prove that {Zk. 1 S k S n} is a martingale. Suppose now all Y k are bounded by a constant A, and define a and M as in the proof of Theorem 9.4.1, with the Xk there replaced by the Sk here. Prove that Thus we obtain an improvement on Exercise 3 of Sec. 5.3. [This is communicated by Doob.]