Definition: For any real number $r > 0$, $\x_n$ converges in $r$th mean to $\x$, denoted $\x_n \overset{r}{\longrightarrow} \x$, if
\[\Ex\norm{\x_n-\x}^r \to 0.\]The most common case is $r=2$, where it is called convergence in quadratic mean and denoted $\x_n \inQM \x.$
See also: Proving consistency via quadratic mean