Considering any $(\mathcal{F}_r)_{r \geq 0}$-adapted processes $X_r,$ if we want to prove the convergence of $X_r$ and depending on the properties of the processes, we often use filtrations other than $\mathcal{F}_r$ $(\mathcal{F}_{r+},\mathcal{F}_{r-},...),$ in order to facilitate the work or to obtain a desired property.
In general, the filtration, does it affect the convergence of $X_r$? How?