If the expectation of the norm of random variable is bounded, does that imply that the expectation of the squared norm is bounded?

Question

Let $\mathbf{x}$ be a random vector. If $\mathbb{E}[\|\mathbf{x}\|]$ is upper bounded, does that imply that $\mathbb{E}[\|\mathbf{x}\|^2]$ is also upper bounded?

The Jensen's inequality goes the other way, but it seems quite intuitive that the proposition holds. If not, can you think of an example where it does not hold?

Thank you

Answer 1

Yeah, take $X$ so that $$ P(X = \sqrt{k}) = \frac{6}{\pi^2} \frac{1}{k^2}.$$

Then $\| X\| = X$, and $$E[X] = \frac{6}{\pi^2}\sum \frac{1}{k^{3/2}} < \infty.$$

Trying to compute $E[X^2]$ gives the harmonic series, though, which is not finite.

Answer 2

This does not hold for the same reason that $\mathbb{E}[|X|]<\infty$ does not imply $\mathbb{E}[X^2]<\infty$. Take any random variable $X$ for which $\mathbb{E}[|X|]<\infty$ but $\mathbb{E}[X^2]=\infty$. Then let the random vector $\textbf{x}$ be $\textbf{x}=(X,0,\dots,0)$. Then $\|\textbf{x}\|=|X|$ but $\|\textbf{x}\|^2=X^2$, so $\mathbb{E}[\|\textbf{x}\|<\infty$ but $\mathbb{E}[\|\textbf{x}\|^2]=\infty$.