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This brief pedagogical note re-proves a simple theorem on the convergence, in$L_2$ and in probability, of time averages of non-stationary time series to themean of expectation values. The basic condition is that the sum of covariancesgrows sub-quadratically with the length of the time series. I make no claim tooriginality; the result is widely, but unevenly, spread bit of folklore amongusers of applied probability. The goal of this note is merely to even out thatdistribution.

A Simple Non-Stationary Mean Ergodic Theorem, with Bonus Weak Law of Large Numbers