When I google the central limit theorem it says the following: The central limit theorem states that if you have a population with mean $μ$ and standard deviation $σ$ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed.
My question is the following: does the central limit theorem only apply to sampling distributions of the sample mean? Or can it apply to any sampling distribution of any statistic? Such as sample variance, sample proportion, or difference between two sample means, etc..
Thank You For Your Time
The Central Limit Theorem does not apply to the sampling distribution of all statistics, not even those involving the sample means.
Here is a counterexample. Let $X_1,\dots ,X_n \sim_{i.i.d.} F_X$ and let $Y_1,\dots, Y_m \sim_{i.i.d.} F_Y$, independent of the $X_i$. Let the statistic be $\hat \theta = \sum_{i=1}^n X_i/n - \sum_{j=1}^m Y_j/m$. Suppose $n\rightarrow \infty$, but $m$ is fixed. Then the distribution of $$\frac{\hat \theta -E(\hat \theta)}{Var^{1/2}(\hat \theta)}$$ does not converge to $N(0,1)$ or any other normal distribution.