Past exam of the martingales and financial mathematics course of the University of Oxford / 2008 / A / 2 / c

The zero means and orthogonality give the exact second momentThus $A_N\to 0$ in $L^2$. Chebyshev's inequality gives the requested weak law of large numbers:Independence is unnecessary; the stated second-moment orthogonality is sufficient.

For an explicit almost surely convergent subsequence take $N_j=2^j$. For each positive integer $m$,The First Borel-Cantelli lemma makes only finitely many such exceedances occur. Intersecting the resulting probability-one events over $m$ gives $A_{2^j}\to 0$ almost surely.