asymptotic normality and central limit theorem

Here's the question

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

• Asymptotic normality is a property, while the CLT as its name suggests is a theorem. This theorem tells us that certain s.v. with certain conditions are asymptotically normally distributed. So it proves that certain s.v. have this asymptotic normality property. – Marc Mar 21 '14 at 7:55
• what does s.v. mean? – natsu Mar 21 '14 at 8:00
• sorry, stochastic variable – Marc Mar 21 '14 at 9:21

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.