Let $\{\xi_n\}_{n\geq1}:(\Omega,\mathcal{F},\mathbb{P})\to(\mathbb{R^1},\text{Bor})$ be a sequence of random variable. What are the connections between the following two types of convergence as $n\to\infty$?
1) $\xi_n\to\xi$ almost surely
2) $\xi_n\to\xi$ in the p-th mean, i.e. $\sum_{n\geq 1}E(|\xi_n-\xi|^p)<\infty$
According to Wikipedia (https://en.wikipedia.org/wiki/Convergence_of_random_variables), convergence in p-th order mean implies convergence in probability, and convergence in probability implies there exists a sub-sequence which almost surely converges. Are there any other connections?
Edit: I realize that the second convergence is not convergence in the mean, so please ignore the above paragraph. What are the connections between just these two types of convergence?