What can you conclude about the first moment of a variable given the 3rd moment exists and is finite

Question

Suppose you are given a random variable $X$ and told that $E[X^3]$ exists and finite. Can you conclude that $E[X]$ exists and is finite? What about $E[X^2]$?

How would you argue rigorously whether this is true?

Answer 1

Yes. In general $E(X^n)< \infty$ implies that $E(X^m)< \infty$ for $m \leq n$.

An outline of proof is as follows:

Without loss of generality assume $X \geq 0.$ Let $f_X(x)$ be the density function of $X$

$E(X^m)=\int_{0}^{\infty}x^mf_X(x)dx=\int_{0}^1x^mf_X+\int_{1}^{\infty}x^mf_X$.

The first integral is bounded by 1, and the second integral converges, as $x^m\leq x^n$ on $[1,\infty]$.