Let X1, X2,... , be independent real random variables with EXn = 0 for all n and Sn := X1 + ··· + Xn . Let Bn be the smallest σ-algebra for which X1,..., Xn are measurable. Then {Sn, Bn } is a martingale (see the example before Proposition 10.3.2).

(a) Give an example where Sn converge a.s., but are not L1-bounded.

(b) If |X j |≤ 2 a.s. for all j , and Sn converges a.s., show that Sn converges in L2. Hint: Use the three-series theorem (9.7.3).