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>[almost-sure convergence] first give <weak convergence> in the <Hilbert space> $L^2$; <weak convergence> together with convergence of <norms> gives <strong convergence>.