# Terminology: “Central Limit Theorem” for Delta Method

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(!).

• 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. – chRrr Feb 26 at 14:32