Suppose $\{M_n^k,n=1,2,\cdots,N\}_{k\in\mathbb N}$ is a sequence of martingale on a fixed probability space. Suppose the terminal distributions $\mu_k=\mathcal L(M_N^k), k=1,2,...$ is a tight sequence. I am wondering if the laws of $\{M_n^k\}$ is tight or not.
In general, I am interested in knowing if there are any general criterions for tightness of laws of discrete time martingales.
Thanks a lot!