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This is a question about when is it appropriate to call an asymptotic normality statement, the "Central Limit Theorem" (CLT).

More specifically, suppose I have $X_1, X_2, \dots X_n$ iid from a distribution with mean $\mu$ and finite variance $\sigma^2$. Let $\bar{X}$ be the sample mean. By the CLT, $$ \sqrt{n}(\bar{X} - \mu) \overset{d}{\to} N(0, \sigma^2)\,. $$ The above equation is a consequence of the CLT, but is also often referred to as a CLT. That is, the asymptotic normality itself is referred to as the CLT. This I understand to be colloquial norm, and I am ok with.

But now, suppose for a differentiable function $g$, I use the Delta Method to obtain an asymptotic statement about $g(\bar{X})$. That is, for some $\lambda^2$, $$ \sqrt{n}(g(\bar{X}) - g(\mu)) \overset{d}{\to} N(0, \lambda^2)\,. $$ The above is also often referred to as a CLT. The theorem however only applies to a sample average. So the asymptotic normality of $g(\bar{X})$ should not be referred to as a CLT. If the above is referred to as a CLT, then I would imagine asymptotic normality of MLEs should can be referred to as a CLT(!).

I am wondering if the community has any strong (or weak) opinions about this?

Edit:

Another example is the asymptotic normality of the median, which is also often referred to as the CLT for a median (or any quantile).

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  • $\begingroup$ The statement about the convergence of $\sqrt{n}(g(\overline{X}) - g(\mu))$ can also be found under the name "Cramér's Theorem", see for example "A Course in Large Sample Theory" (Thomas Ferguson, 1996), Theorem 7 (Cramer). It needs the distribution of $\sqrt{n}(\overline{X}-\mu)$ which will often be given by the CTL. Strictly speaking the CTL is only a result on the asymptotic distribution of the sample mean, NOT of functions of sample means. $\endgroup$ – chRrr Feb 26 at 14:32
  • $\begingroup$ @chRrr Can't say I disagree. $\endgroup$ – Greenparker Feb 28 at 11:10

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