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> it seems like using methods that rely on stationary distributions still treat the data, after pre-processing, as i.i.d, rather than predicting values from their correlated history.

Not sure what you mean exactly. Stationarity in time series is not i.i.d. The whole point of ARIMA modelling is to model, after transforming to stationarity, the remaining temporal correlation. ARMA is just a limited-parameter fit of the autocorrelation function.




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