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?


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

  • $\begingroup$ 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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