Expectation of norm of a random variable Let $x_k$ be a random vector such that its expectation
$$
E[\Vert x_k \Vert]&lt;a
$$
for some $a&gt;0$. Then can we say that
$$
E[\Vert x_k \Vert^2]&lt;a^2 ?
$$
$\mathbb{E}[\lVert X\rVert^2]$ 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 $\mathbb{E}[X] = \sum_{n=1}^\infty \frac{1}{\zeta(3)}\cdot\frac{1}{n^2} = \frac{\pi^2}{6\zeta(3)}$)
No, not in general, but you can say that $\left[\mathbb{E}\left(\biggr|\biggr|\vec{X_k}\biggr|\biggr|\right)\right]^2 &lt;a^2$ .