asymptotic normality and central limit theorem

Question

Here's the question

Can somebody explain the difference between asymptotic normality and central limit theorem? They seem very similar to me.

Answer 1

Asymptotic normality is a feature of a sequence of probability distributions. We say that a sequence of probability distributions is asymptotically normal if it converges weakly to the normal distribution.

Asymptotic normality and the central limit theorem are closely related notions.

The central limit theorem gives an example of a sequence that is asymptotically normal. It establishes that probability distributions corresponding to the sequence of random variables $$ Y_n=\frac1{\sqrt n}\sum_{i=1}^nX_n, $$ where $X_1,\ldots,X_n$ are iid random variables with $\operatorname EX=0$ and $\operatorname EX^2=1$, converges weakly to the standard normal distribution. The sequence of probability distributions of $Y_n$ is asymptotically normal.

The term asymptotic normality is usually used in statistics to describe asymptotic properties of an estimator.

Wikipedia has some nice pages about these topics (see Asymptotic distribution and Central limit theorem ).