$L^p$ convergence for real valued random vectors is defined as \begin{align*} \mathbb{E}[\|X_n-X\|^p]\underset{ n\uparrow \infty}{\rightarrow}0, p\geq 1. \end{align*} What norm is typically used in this definition for vectors?
And does this convergence imply convergence of all the moments? This appears to address the question but seems to implicitly define $L^p$ convergence via the $p$-norm, \begin{align*} \mathbb{E}[\|X_n-X\|_p]\underset{ n\uparrow \infty}{\rightarrow}0, p\geq 1. \end{align*} instead of a $p$th power of a norm, as above.