Does finite k-th central moment imply finite k-th raw moment?

Question

Does the existence of a finite $k$-th central moment $\mathbb{E}(|(X- \mathbb{E}X)^k|) < \infty$ imply the existence of a finite $k$-th raw moment $\mathbb{E}(|X^k|) < \infty$? If so how can I prove it?

I know only that the opposite is true because we can easily write any $k$-th central moment as a linear combination of raw moments with orders $k, k-1, ...,1$.

Answer 1

Yes. The idea is the same as the one you're comfortable with; use the relation $$|X|^k = |(X - \mu) + \mu |^k$$ and expand the right side with the binomial theorem to express the $k$-th raw moment as a linear combination of the lower-order central moments.

Answer 2

By the triangle inequality for $L^p$ spaces ($p\geq 1$) $$ \lVert X \rVert_k\leq \lVert X-EX \rVert_k+\lVert EX \rVert_k<\infty $$ for $k\geq 1$. Here $$ \lVert X \rVert_k=(E|X|^k)^{1/k}. $$