Expectation of norm of a random variable

Question

Let $x_k$ be a random vector such that its expectation $$ E[\Vert x_k \Vert]<a $$ for some $a>0$. Then can we say that $$ E[\Vert x_k \Vert^2]<a^2 ? $$

Answer 1

$\mathbb{E}[\lVert X\rVert^2]$ need not even be defined in general. Consider the random variable on $\mathbb{N}^\ast$ (vector of dimension 1) with probability mass function $$ \mathbb{P}\{ X = n\} = \frac{1}{\zeta(3)}\cdot\frac{1}{n^3}. $$ (for which you do have $\mathbb{E}[X] = \sum_{n=1}^\infty \frac{1}{\zeta(3)}\cdot\frac{1}{n^2} = \frac{\pi^2}{6\zeta(3)}$)

Answer 2

No, not in general, but you can say that $\left[\mathbb{E}\left(\biggr|\biggr|\vec{X_k}\biggr|\biggr|\right)\right]^2 <a^2$ .