Hints: Partial sums of independent stochastic variables

Here it is important to be familiar with the equivalent definitions of a martingale, where the definition in \((2.3)\) in ABG in particular is the easiest to work with. Recall also the definition of the history \(\mathcal{F}_{m}\), and what conditioning events on this implies with respect to whether something is known or potentially independent (see section 2.1.1 in ABG for details). For the transformation in (b), can you reduce the problem to (a)?