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  • $\begingroup$ You seem to be asserting a law of large numbers rather than the CLT. $\endgroup$ Commented Mar 14, 2019 at 14:59
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    $\begingroup$ I am not sure why you would say this, @whuber. The above give an intuitive proof that $E[f((X_1+...+X_n)/\sqrt n)]$ converges to $E[f(Z)]$ where $Z\sim N(0,1)$ for a large class of functions $f$. This is the CLT. $\endgroup$ Commented Mar 14, 2019 at 16:09
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    $\begingroup$ I see what you mean. What gives me pause is that your assertion concerns only expectations and not distributions, whereas the CLT draws conclusions about a limiting distribution. The equivalence between the two might not immediately be evident to many. Might I suggest, then, that you provide an explicit connection between your statement and the usual statements of the CLT in terms of limiting distributions? (+1 by the way: thank you for elaborating this argument.) $\endgroup$ Commented Mar 14, 2019 at 18:23
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    $\begingroup$ Really great answer. I find this much more intuitive than characteristic function kung-fu. $\endgroup$ Commented Mar 21, 2020 at 23:49
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    $\begingroup$ For anyone seeing this - this is also known as the Lindeberg swapping argument! $\endgroup$ Commented May 23, 2024 at 4:28