Let $X_n$ be a real-valued martingale. Can one proof this: $\mathbb{E}((X_{n+m-1}-X_{n})(X_{n+m}-X_{n+m-1})^2|\mathcal{F}_n) \neq 0$ and $\mathbb{E}((X_{n+m-1}-X_{n})^2(X_{n+m}-X_{n+m-1})|\mathcal{F}_n) = 0$?
I am trying to understand the proof of proposition 1 on page 45 of this paper.
The second part is true (provided that all the expectations are well defined). Let $U:= (X_{n+m-1}-X_n)^2$ and $V=X_{n+m}-X_{n+m-1}$. Then by the tower property of the condition expectation, $$ \mathbb E\left[UV\mid \mathcal F_n\right]=\mathbb E\left[\mathbb E\left[ UV\mid\mathcal F_{n+m-1}\right]\mid \mathcal F_n\right], $$ Since $U$ is $\mathcal F_{n+m-1}$-measurable, it follows that $$ \mathbb E\left[ UV\mid\mathcal F_{n+m-1}\right]= U\mathbb E\left[ V\mid\mathcal F_{n+m-1}\right] $$ and by the martingale property, the last conditional expectation is $0$.