Convergence in density of a sequence

A sequence $a_n$ converges in density to $a$ when, for every $\epsilon >0$,

The displayed condition implies convergence in density of a sequence to $\ell$, since the fraction of indices with $|a_n-\ell |\ge \epsilon$ is at most ${\epsilon }^{-2}$ times the averaged squared error. A signed Cesaro limit alone is insufficient: $a_n=(-1{)}^n$ has zero signed average but no density limit.

A sequence $a_n$ converges in the Cesaro sense to $a$ when $N^{-1}{\sum }_{n=1}^N{a}_n\to a$. For bounded sequences, convergence in density to $a$ is equivalent to Cesaro convergence of $|a_n-a|$ to zero.