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Given a sample path, we can roughly tell whether the mean changes over the time, and, when it doesn't, whether the deviation from mean changes over the time. (Correct me if I am wrong.)

But that is not enough for telling if the process is stationary (here, stationarity is defined in terms of the first and the second moments), as we still need to check the sample covariance for every time lag. So can we do that by directly looking at a sample path without estimating the covariance?

Thanks!

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Since for covariance stationary processes the covariance does not depend on t but on h for $\text{Cov}[y_t,y_{t+h}]$, it is possible to detect serial pattern in a time series. Consider quarterly measured GNP. Here, one can usually detect an increasing GNP in summer and an decreasing GNP in winter. Hence, there is dependence depending on the season of the year.

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    $\begingroup$ Could you explain how this has a bearing on the question? The "serial pattern" you describe appears to concern the mean and that situation is already covered in the question. The focus of your answer should be on the covariance, which in fact can be stationary for a seasonal process. $\endgroup$
    – whuber ♦
    Commented Feb 15, 2014 at 22:05

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