Consider the following important theorem, due to Jacod (1997), on stable convergence of triangular arrays of random variables. In what follows $\mathcal{F}_t$ indicates a filtration on $[0,1]$ and $t_{j,n}$ is a partition of the time-interval $[0,1]$.
My question: is it strictly necessary that the conditioning that appears in conditions from $1)$ to $5)$ is taken with respect to $\mathcal{F}_{t_{j-1,n}}$ ? Would the theorem still apply if, for example, conditioning with respect to $\mathcal{F}_{t_{j-2,n}}$ is considered?
Here are some ideas, for example for assumption 1. Let $$ \tag{1'}\sum_{j=1}^n\mathbb E\left[Y_{t_{j,n}}\mid\mathcal F_{t_{j-2},n}\right]\to 0 \mbox{ in probability}. $$ If (1') holds, in order to check (1), we have to prove that $$ \tag{1'}\sum_{j=1}^nd_{n,j}\to 0 \mbox{ in probability}, $$ where $d_{n,j}=\mathbb E\left[Y_{t_{j,n}}\mid\mathcal F_{t_{j-1},n}\right]-\mathbb E\left[Y_{t_{j,n}}\mid\mathcal F_{t_{j-2},n}\right]$. The fact that $\left(d_{n,j}\right)$ is a martingale difference sequence can help.