A type of convergence of random variable

Question

Let $\{\xi_n\}_{n\geq1}:(\Omega,\mathcal{F},\mathbb{P})\to(\mathbb{R^1},\text{Bor})$ be a sequence of random variable, and $$\sum_{n\geq 1}E(|\xi_n-\xi|^p)<\infty.$$ Is there a name for this type of convergence? What are the connections between this type of convergence and $L^p$ convergence?