Ergodic for the mean

•Stationary time series.•Consistent in mean square.•Sucient and necessary condition.•Asymptotic average uncorrelatedness.

We discuss ergodicity for the mean in the sense that the sample average converges in mean square to the population mean of a stationary stochastic process. This differs from ergodicity in a measure theoretic sense. It is widely known that asymptotic uncorrelatedness is sufficient for ergodicity for the mean. We weaken this assumption to “asymptotic average uncorrelatedness” and show that it cannot be further weakened: Our condition is necessary and sufficient.