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@NateEldredge Thanks for you comment. I edited my question. I was also trying to prove uniform integrability. All we know is $\lim_{A\to\infty}\sup_{n}E_{Q_n}E[S_k\mathbf1_{Sk>A}]=0$ for all $k=0,…,N−1$. This is stated in the same proof before equation (3.17). But this estimate is used to prove tightness of the $\{Q_n\}$
