Does finite fourth moment imply finite third absolute moment?

Question

This may be a silly question, but, for a random variable $X$ defined on reals, I am wondering if the existence of the finite fourth central moment $E[(X-E[X])^4]$ implies the existence of the finite third absolute moment $E[|X-E[X]|^3]$? Formally, does the following statement hold:

$$E[(X-E[X])^4]<\infty\Rightarrow E[|X-E[X]|^3]<\infty$$

The reason for my question is that, while trying to apply the Berry-Esseen Theorem to bound the total variation distance between the distribution of the (appropriately normalized) mean of $n$ i.i.d. random variables and the standard normal distribution, I am finding that the fourth central moment of the i.i.d. random variables the distribution of whose average is being approximated is much easier to compute than their third absolute moment...

Answer 1

Yes.

Proof: Apply the inequality $|x|^3\leqslant1+x^4$ to $x=X-E[X]$ and integrate.

Answer 2

Split the random variable into the following regions

1. $ |X - E[X]| > 1 $. In this region, $|X-E[X]|^3 < |X-E[X]|^4$

2. $|X - E[X]| \leq 1$. In this region, $|X-E[X]|^3 \leq 1 $.

Hence conclude that $E\left[|X-E[X]|^3\right] < E\left[|X-E[X]|^4\right] + 1$