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 
is a martingale.
To show this we need to show that (1) 
![\displaystyle E(\theta_{k+1}|\mathcal{D}_{k}) \\ = E\left[\sum_{l=1}^{k+1}\left(\eta_{l}-E(\eta_{l}|\mathcal{D}_{l-1})\right)|\mathcal{D}_{k}\right] \\ = \sum_{l=1}^{k+1}E\left[\eta_{l}-E(\eta_{l}|\mathcal{D}_{l-1})|\mathcal{D}_{k}\right] \\ = E(\eta_{k+1}|\mathcal{D}_{k})-E\left[E(\eta_{k+1}|\mathcal{D}_{k})|\mathcal{D}_{k}\right]+\sum_{l=1}^{k}E(\eta_{l}|\mathcal{D}_{k})-E\left[E(\eta_{l}|\mathcal{D}_{l-1})|\mathcal{D}_{k}\right] \\ = E(\eta_{k+1}|\mathcal{D}_{k})-E\left[E(\eta_{k+1}|\mathcal{D}_{k})|\mathcal{D}_{k}\right]+\sum_{l=1}^{k}\eta_{l}-E\left[E(\eta_{l}|\mathcal{D}_{l-1})|\mathcal{D}_{k}\right] \\ \mbox{ since }\eta_{l}\mbox{ is }\mathcal{D}_{i\ge l}\mbox{-measurable} \\ = E(\eta_{k+1}|\mathcal{D}_{k})-E\left[E(\eta_{k+1}|\mathcal{D}_{k})|\mathcal{D}_{k}\right]+\sum_{l=1}^{k}\eta_{l}-E(\eta_{l}|\mathcal{D}_{l-1}) \\ \mbox{ since }\mathcal{D}_{l-1}\le\mathcal{D}_{k}\mbox{ and }E(\eta_{l}|\mathcal{D}_{l-1})\mbox{ is }\mathcal{D}_{l-1}\mbox{-measurable} \\ = E(\eta_{k+1}|\mathcal{D}_{k})-E(\eta_{k+1}|\mathcal{D}_{k})+\sum_{l=1}^{k}\eta_{l}-E(\eta_{l}|\mathcal{D}_{l-1}) \\ \mbox{ since }E(\eta_{k+1}|\mathcal{D}_{k})\mbox{ is }\mathcal{D}_{k}\mbox{-measurable} \\ = \theta_{k}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++E%28%5Ctheta_%7Bk%2B1%7D%7C%5Cmathcal%7BD%7D_%7Bk%7D%29%09%5C%5C+%3D+E%5Cleft%5B%5Csum_%7Bl%3D1%7D%5E%7Bk%2B1%7D%5Cleft%28%5Ceta_%7Bl%7D-E%28%5Ceta_%7Bl%7D%7C%5Cmathcal%7BD%7D_%7Bl-1%7D%29%5Cright%29%7C%5Cmathcal%7BD%7D_%7Bk%7D%5Cright%5D+%5C%5C+%3D+%5Csum_%7Bl%3D1%7D%5E%7Bk%2B1%7DE%5Cleft%5B%5Ceta_%7Bl%7D-E%28%5Ceta_%7Bl%7D%7C%5Cmathcal%7BD%7D_%7Bl-1%7D%29%7C%5Cmathcal%7BD%7D_%7Bk%7D%5Cright%5D+%5C%5C+%3D+E%28%5Ceta_%7Bk%2B1%7D%7C%5Cmathcal%7BD%7D_%7Bk%7D%29-E%5Cleft%5BE%28%5Ceta_%7Bk%2B1%7D%7C%5Cmathcal%7BD%7D_%7Bk%7D%29%7C%5Cmathcal%7BD%7D_%7Bk%7D%5Cright%5D%2B%5Csum_%7Bl%3D1%7D%5E%7Bk%7DE%28%5Ceta_%7Bl%7D%7C%5Cmathcal%7BD%7D_%7Bk%7D%29-E%5Cleft%5BE%28%5Ceta_%7Bl%7D%7C%5Cmathcal%7BD%7D_%7Bl-1%7D%29%7C%5Cmathcal%7BD%7D_%7Bk%7D%5Cright%5D+%5C%5C+%3D+E%28%5Ceta_%7Bk%2B1%7D%7C%5Cmathcal%7BD%7D_%7Bk%7D%29-E%5Cleft%5BE%28%5Ceta_%7Bk%2B1%7D%7C%5Cmathcal%7BD%7D_%7Bk%7D%29%7C%5Cmathcal%7BD%7D_%7Bk%7D%5Cright%5D%2B%5Csum_%7Bl%3D1%7D%5E%7Bk%7D%5Ceta_%7Bl%7D-E%5Cleft%5BE%28%5Ceta_%7Bl%7D%7C%5Cmathcal%7BD%7D_%7Bl-1%7D%29%7C%5Cmathcal%7BD%7D_%7Bk%7D%5Cright%5D+%5C%5C+%5Cmbox%7B+since+%7D%5Ceta_%7Bl%7D%5Cmbox%7B+is+%7D%5Cmathcal%7BD%7D_%7Bi%5Cge+l%7D%5Cmbox%7B-measurable%7D+%5C%5C+%3D+E%28%5Ceta_%7Bk%2B1%7D%7C%5Cmathcal%7BD%7D_%7Bk%7D%29-E%5Cleft%5BE%28%5Ceta_%7Bk%2B1%7D%7C%5Cmathcal%7BD%7D_%7Bk%7D%29%7C%5Cmathcal%7BD%7D_%7Bk%7D%5Cright%5D%2B%5Csum_%7Bl%3D1%7D%5E%7Bk%7D%5Ceta_%7Bl%7D-E%28%5Ceta_%7Bl%7D%7C%5Cmathcal%7BD%7D_%7Bl-1%7D%29+%5C%5C+%5Cmbox%7B+since+%7D%5Cmathcal%7BD%7D_%7Bl-1%7D%5Cle%5Cmathcal%7BD%7D_%7Bk%7D%5Cmbox%7B+and+%7DE%28%5Ceta_%7Bl%7D%7C%5Cmathcal%7BD%7D_%7Bl-1%7D%29%5Cmbox%7B+is+%7D%5Cmathcal%7BD%7D_%7Bl-1%7D%5Cmbox%7B-measurable%7D+%5C%5C+%3D+E%28%5Ceta_%7Bk%2B1%7D%7C%5Cmathcal%7BD%7D_%7Bk%7D%29-E%28%5Ceta_%7Bk%2B1%7D%7C%5Cmathcal%7BD%7D_%7Bk%7D%29%2B%5Csum_%7Bl%3D1%7D%5E%7Bk%7D%5Ceta_%7Bl%7D-E%28%5Ceta_%7Bl%7D%7C%5Cmathcal%7BD%7D_%7Bl-1%7D%29+%5C%5C+%5Cmbox%7B+since+%7DE%28%5Ceta_%7Bk%2B1%7D%7C%5Cmathcal%7BD%7D_%7Bk%7D%29%5Cmbox%7B+is+%7D%5Cmathcal%7BD%7D_%7Bk%7D%5Cmbox%7B-measurable%7D+%5C%5C+%3D+%5Ctheta_%7Bk%7D+&bg=ffffff&fg=333333&s=0&c=20201002 "\displaystyle E(\theta_{k+1}|\mathcal{D}_{k}) \\ = E\left[\sum_{l=1}^{k+1}\left(\eta_{l}-E(\eta_{l}|\mathcal{D}_{l-1})\right)|\mathcal{D}_{k}\right] \\ = \sum_{l=1}^{k+1}E\left[\eta_{l}-E(\eta_{l}|\mathcal{D}_{l-1})|\mathcal{D}_{k}\right] \\ = E(\eta_{k+1}|\mathcal{D}_{k})-E\left[E(\eta_{k+1}|\mathcal{D}_{k})|\mathcal{D}_{k}\right]+\sum_{l=1}^{k}E(\eta_{l}|\mathcal{D}_{k})-E\left[E(\eta_{l}|\mathcal{D}_{l-1})|\mathcal{D}_{k}\right] \\ = E(\eta_{k+1}|\mathcal{D}_{k})-E\left[E(\eta_{k+1}|\mathcal{D}_{k})|\mathcal{D}_{k}\right]+\sum_{l=1}^{k}\eta_{l}-E\left[E(\eta_{l}|\mathcal{D}_{l-1})|\mathcal{D}_{k}\right] \\ \mbox{ since }\eta_{l}\mbox{ is }\mathcal{D}_{i\ge l}\mbox{-measurable} \\ = E(\eta_{k+1}|\mathcal{D}_{k})-E\left[E(\eta_{k+1}|\mathcal{D}_{k})|\mathcal{D}_{k}\right]+\sum_{l=1}^{k}\eta_{l}-E(\eta_{l}|\mathcal{D}_{l-1}) \\ \mbox{ since }\mathcal{D}_{l-1}\le\mathcal{D}_{k}\mbox{ and }E(\eta_{l}|\mathcal{D}_{l-1})\mbox{ is }\mathcal{D}_{l-1}\mbox{-measurable} \\ = E(\eta_{k+1}|\mathcal{D}_{k})-E(\eta_{k+1}|\mathcal{D}_{k})+\sum_{l=1}^{k}\eta_{l}-E(\eta_{l}|\mathcal{D}_{l-1}) \\ \mbox{ since }E(\eta_{k+1}|\mathcal{D}_{k})\mbox{ is }\mathcal{D}_{k}\mbox{-measurable} \\ = \theta_{k}")
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