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)
.