Let $\{X_{i}\}_{i=1}^{n}$ and $\{Y_{i}\}_{i=1}^{n}$ are two sequences of random variables such that $\bar{X} = \sum_{i=1}^{n} X_{i}$ and $\bar{Y} = \sum_{i=1}^{n} Y_{i}$ asymptotically converges in distribution to $\mathcal{N}(0,\sigma)$, where $\sigma >0$. If $X_{i}$ and $Y_{j}$ are not independent for $1\leq i,j \leq n$, then, is it possible that $\bar{X}$ and $\bar{Y}$ will become asymptotically independent under some condition. If yes, then under what condition?
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1$\begingroup$ It depends on what you mean by "converges." The commonest senses are surely; almost surely; in probability; and in distribution. In your setting usually the latter is the intended sense (as in the Central Limit Theorem), but then your question makes little sense. What sense of convergence, then, do you have in mind? $\endgroup$– whuber ♦Sep 10, 2022 at 19:03
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$\begingroup$ In the above question, we are talking about convergence in distribution. $\endgroup$– BhishamSep 10, 2022 at 19:08
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2$\begingroup$ Re the edit: as I mentioned, for convergence in distribution independence has no meaning. If you wish to discuss such things, you need to consider the convergence of the vector random variable $(X_i,Y_i).$ $\endgroup$– whuber ♦Sep 10, 2022 at 19:09
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