I am trying to understand the answer to questions here and square it with what I know to be true. In particular, the result there is that for a centered random variable $X$
$$ E\left[\left( \frac{1}{\sqrt{n}} \sum_{i=1}^n X_i\right)^p\right] \leq C_p E[|X|^p] $$
for some constant $C_p$. I know that this bound is sharp (by, for example, the Marcinkiewicz–Zygmund inequality). However, this means that if $X$ does not have a finite $p$ moment, the $p$ moment of the sample mean is infinite. But then, I know that
$$ \frac{1}{\sqrt{n}} \sum_{i=1}^n X_i $$
converges to a standard normal random variable as $n \to \infty$ by the Central Limit Theorem which should have finite moments for any $p$. In fact for $n$ large enough, the previous average is approximately normal. How do I square these two facts? Are the $p$ moments of a sample mean finite or not in this case?