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Since the book says, it will use time series to mean either realization of a process or a process, I have no idea how to interpret the following sentence.

"This notion, called weak stationary(i.e.autocovariance is independent of time but only depends upon time separation of variable), when the mean is constant, is fundamental in allowing us to analyze sample time series data when only one series is available."

  1. How do I analyze the data if only one series is given? In particular, it could mean one stochastic process which makes sense. However for a specific realization say $(x_{t_i})$ is measured data points at $t_1<t_2<t_3<\dots$, I do not see it makes sense to me. There is no distribution associated to say $x_{t_1}$ as I have only one point. I guess the interpretation is several realizations of the process being required.

  2. Why mean is constant is important here?

Reference: Stoffer and Shumway. Time Series Analysis and Its Applications, paragraph right before Sec 1.4 on pg 19

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If we know the probability distribution of data generation, we can calculate unconditional quantities directly without the help of observations. But we don't know the distribution and we have only one instantiation of time series in real life.

They seek to find a way to calculate unconditional quantities from a time average over the time series observed. That is to say, we replace the ensemble average over different time series instantiations by the time average of one time series.

And the properties of time series in which two quantities agree are called ergodicity. Loosely speaking, ergodicity is satisfied in case of weak stationary time series.

Let's consider our task is to calculate the mean of the distribution. If the mean of series is not constant, the time average would not agree with ensemble average at a point of time. Because the mean changes all the time and time average will give a different number depending on the time interval which is averaged.

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