Mean-square criterion for convergence in density
= Mean-square criterion for convergence in density
{title2=$\frac1N\sum_{n<N}|a_n-\ell|^2\to0$}
The displayed condition implies <convergence in density of a sequence> to $\ell$, since the fraction of indices with $|a_n-\ell|\geq\varepsilon$ is at most $\varepsilon^{-2}$ times the averaged squared error. A signed <Cesaro limit> alone is insufficient: $a_n=(-1)^n$ has zero signed average but no density limit.