We have often seen that in predictive modeling, we need to make certain important but limiting assumptions in order to build practical models. With time series models, one of the most common assumptions to make that render the modeling task significantly simpler is the stationarity assumption.

Stationarity essentially describes that the probabilistic behavior of a time series does not change with the passage of time. There are two versions of the stationarity property that are commonly used. A stochastic process is said to be strictly stationary when the joint probability distribution of a sequence of points starting at time t, Y_t, Y_t+1, ..., Y_t+n, is the same as the joint probability distribution of another sequence of points starting at a different time T, Y_T, Y_T+1, ..., Y_T+n.

To be strictly stationary, this property must hold for any choice of time t and T, and for any sequence length n. In particular, because we can choose n = 1, this means that the probability distributions...