Example of a random variable for which $\|\mathbb{E}[x]\| <\infty$ but $\mathbb{E}[\|x\|]=\infty$

Question

We know by Jensen's inequality and the convexity of the euclidean norm that,

$$\mathbb{E}[\|x\|]\geq\|\mathbb{E}[x]\|$$

I wonder, then, if it's possible to have a random variable $x$ whose expected norm is finite but the norm of the expectation is infinite?

Answer 1

"I wonder, then, if it's possible to have a random variable $x$ whose expected norm is finite but the norm of the expectation is infinite?": The inequality you cite says precisely that this is impossible.

The sensible question is the other way around. In fact, by the standard definitions, if $\Bbb E[||x||]=\infty$ then $\Bbb E[x]$ is undefined, so the question of whether it can have finite norm is meaningless.