For which parameters does the Central Limit Theorem work?

I have a question about the Central Limit Theorem in the context of estimating a population parameter through a sample statistic.

The most known case is that the CLT asserts that a sampling distribution of the means will become approximately a normal distribution for a large sample size. I saw this also works for the sampling distribution of the proportion.

As I understood the CLT doesn't generally guarantee the sampling distibution to be normal for arbitrary sample statistics. Are there any other sample statistics for which the CLT does so, and if so, for which?

• At its original form, CLT is for the sum of a sequence of random variables (or more generally, a triangular array). It is a pure probabilistic result which could have nothing to do with parameters. Therefore, do you actually want to ask except sample mean, does CLT establish normal convergence result for other sample statistics, such as proportion? Dec 22, 2017 at 20:16
• the proportion is in fact a sample mean. Dec 22, 2017 at 20:27
• The CLT would apply to any quantity which can be written as a mean and for which the other conditions of the CLT hold (whichever version of it you're looking at). For example, you could write a version of the CLT in terms of the standardized second raw sample moment, $\overline{X^2}$, simply by the expedient of starting with "Let $Y_i=X^2_i$" and applying the CLT to the $Y_i$ values. Dec 22, 2017 at 22:50