Convergence in L2 ($X_n\to X\text{ in }{L}^2$)

Convergence in $L^2$ means $\mathbb{E}[(X_n-X{)}^2]\to 0$. It implies convergence in probability by Markov inequality.

If square-integrable random variables satisfy $X_n\to X$ almost surely, then $X_n\to X$ in $L^2$ exactly when $\mathbb{E}[X_n^2]\to \mathbb{E}[X^2]$. Boundedness in $L^2$ and almost-sure convergence first give weak convergence in the Hilbert space $L^2$; weak convergence together with convergence of norms gives strong convergence.