I am still unable to follow the martingales based proof for the Ballot Theorem, so I’ll work out a few more problems first. A problem asks the following. Let be a sequence of decompositions with , and let be -measurable variables. Show that the sequence with
is a martingale.
To show this we need to show that (1) is -measurable. This is clear because takes on a single value (the expectation) conditioned on each . Next, we need to show that (2) .