Power law distributions are defined as follows:
$$\bar{F}(x) = \Pr(X > x) \propto x^{-a}$$
When $0 \lt a \leq 2$, it's called a heavy-tailed distribution. When $a > 2$ it's still a power law distribution, but not a heavy-tailed distribution. When it is a power law, i.e. $0 < a \leq 2$, the mean and the variance are infinite.
My question is what happens to the mean and variance when $a$ is greater than $2$? If the variance is finite when $a\gt 2$, does central limit theorem holds for power-law distributions?
If $X$ is a positive random variable then $EX^{n}=\int_0^{\infty} nx^{n-1}P\{X>t\}\, dt$. The mean if finite if $a>2$. The variance is finite if $a>3$. Yes, you can apply CLT when the variance is finite. For $2<a\leq 3$ you cannot use CLT because the variance is infinity.