Consider a sequence of martingales that are right-continuous with left limits, denoted by $(X^n_t)_{0\le t\le 1}$, such that for each $n\ge 2$,
\begin{eqnarray} (1) && X^n_0=0 \mbox{ and } \sup_{0\le t\le 1}|X^n_t|\le 1; \\ (2) && X^n \mbox{ is contant on } [(k-1)/n,~ k/n) \mbox{ for } 1\le k\le n, \mbox{ i.e. } X^n_t=X^n_{(k-1)/n},~ \forall (k-1)/n\le t<k/n; \\ (3) && \mathbb E[|X^n_{1/n}|]\ge 1/2 \mbox{ and } \mathbb E[|X^n_{2/n}-X^n_{1/n}|]\ge 1;\\ (4)&& \lim_{n\to\infty} \mathbb E[|X^n_{k/n}-X^n_{(k-1)/n}|]=0 \mbox{ for all } k\ge 3. \end{eqnarray}
Roughly speaking, $X^n$ exhibits some "asymptotic continuity" in average on $[\delta, 1]$ for every $\delta>0$, but around zero there is always some jump. My question is whether we can show the tightness of $(X^n)_{n\ge 1}$ (in the Skorokhod sense)? If not, can we prove that $(X^n)_{n\ge 1}$ does not admit any convergent subsequence?
Any answers or comments are highly appreciated.
