Almost-sure convergence and convergence of second moments
= Almost-sure convergence and convergence of second moments
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.