Statistical convergence

Summary

A sequence converges statistically to $L$ when, for every $\varepsilon > 0$, the indices $n$ with $\abs{x_n - L} \ge \varepsilon$ form a set of natural density zero. So the sequence may stray from $L$ infinitely often, but only on a sparse set of indices. Every convergent sequence converges statistically to its limit, but not conversely. For bounded sequences, statistical convergence to $L$ is equivalent to strong Cesàro summability to $L$ with exponent $1$.