1.12. Let {~n, n E N} be a sequence of independent and identically distributed r.v.'s with zero mean and unit variance; and Sn = '2:'j=1 ~j.

Then for any optional LV. a relative to {~n} such that rF(fo) < 00, we have {(ISa!) < ,j2cf(J(X) and J(Sa) = O. This is an extension of Wald's equation due to Louis Gordon. [HINT: Truncate a and put rJk = (S~ljk)

(SLl/~)' then 00 00 l{S;lh} = b ( md:7Y <

b·0?{a > k}/Jk < 2cf(h);

now use Schwarz's inequality followed by Fatou's lemma.]

The next two problems are meant to give an idea of the passage from discrete parameter martingale theory to the continuous parameter theory.