I have the following problem. There are two martingales (they are not, in general, independent), say $S_{i,T}(m)=\sum_{t=m+1}^{m+T}e_{i,t}$ and $S_{j,T}(m)=\sum_{t=m+1}^{m+T}e_{j,t}$.
The innovations/martingale differences $e_{i,t}$ and $e_{j,t}$, as I said, are not necessarily independent but I can assume they have up to the 4-th moment.
In order to apply a theorem which I found, I need to evaluate
$E|E[S_{i,T}(m)S_{j,T}(m)|F_{m}]-E[S_{i,T}(m)S_{j,T}(m)]|$
where $F_{m}$ is the sigma algebra generated by ${e_{i,1},...,e_{i,m}}$ and ${e_{j,1},...,e_{j,m}}$.
The theorem which I need to use requires that
$E|E[S_{i,T}(m)S_{j,T}(m)|F_{m}]-E[S_{i,T}(m)S_{j,T}(m)]| \leq T^{1-\theta}$,
for some $\theta>0$.
Interestingly, I know of a proof of the above if the innovations $e_{i,t}$ and $e_{j,t}$ are mixingales (which is therefore good for the martingale case), but I was wondering whether in the martingale case there is a quick and easy argument to prove the result (or at least an alternative).