Sufficient Conditions for the Central Limit Theorem My understanding is that the central limit theorem applies as long as the variance of the random variable is less than infinity. Is this equivalent to saying that all moments are finite? If not, what is an example of a random variable where its variance is less than infinity, so the central limit theorem applies, but its other moments are infinite?
 A: Finite variance does not imply finite higher moments: A simple illustration can be done with the Pareto distribution, which has tails that obey the power law.  Let $X \sim \text{Pareto}(1, \alpha)$ be a unit-scale Pareto random variable, which has density function:
$$f_X(x) = \alpha x^{-\alpha-1} \quad \quad \quad \text{for all }x \geqslant 1.$$
The $k$th raw moment of this distribution is:
$$\begin{equation} \begin{aligned}
\mathbb{E}(X^k) = \int \limits_\mathscr{X} x^k f_X(x) dx 
&= \int \limits_1^\infty x^k \alpha x^{-\alpha-1} dx \\[6pt]
&= \alpha \int \limits_1^\infty x^{k-\alpha-1} dx \\[6pt]
&= \begin{cases}
\infty & & \text{if } \alpha \leqslant k, \\[6pt]
\frac{1}{\alpha-k} & & \text{if } \alpha > k. \\[6pt]
\end{cases}
\end{aligned} \end{equation}$$
We can see here that the raw moments of this distribution exist (i.e., are finite) for all $k < \alpha$.  Hence, taking $2 < \alpha \leqslant k$ for some integer $k \geqslant 3$, the variance exists but the $k$th central moment does not exist.
