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

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 the mean of expectation values. The basic condition is that the sum of covariances grows sub-quadratically with the length of the time series. I make no claim to originality; the result is widely, but unevenly, spread bit of folklore among users of applied probability. The goal of this note is merely to even out that distribution.