This question came up in a discussion about statistical tests in natural language processing (I'm happy to give you the full story if you are interested).
Suppose we have random variables $X_1, X_2, \dots$ and $Y_1, Y_2, \dots$ All $X_i$ and $Y_i$ are Bernoulli distributed n with the same success probability $p$. The $X_i$ are independent, and the $Y_i$ are independent, but there may be dependencies between the $X_i$ and $Y_i.$
Define the averages $\overline{X}^n = \frac1n \sum_{i=1}^n X_i$, $\overline{Y}^n = \frac1n \sum_{i=1}^n Y_i$ and the $t$-statistics $$ S^n = \sqrt{n} \frac{\overline{X}^n - p}{\sqrt{p(1-p)}}, \qquad T^n = \sqrt{n} \frac{\overline{Y}^n - p}{\sqrt{p(1-p)}}. $$ The central limit theorem guarantees that $S^n$ and $T^n$ converge in distribution to random variables $S, T \sim \mathcal{N}(0,1).$
Suppose that there is a convergent sequence $(c_n)_n$ such that $\text{Cov}(S^n, T^n) = c_n$. Is it true that $$ \text{Cov}(S,T) = \lim_n c_n ? $$
I tried to apply dominated convergence, but failed to find a suitable dominating random variable.