Can stationary time series contain regulary cycles and periods with different fluctuations I just started trying to undestand the notion of stationary in time series. Basically I have 2 questions:

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*Can stationary time series contain regulary cycles and thus seasonality patterns? For exmaple in this tutorial it is stated that stationary time series can not have seasonal components (predicitable cycles) https://otexts.com/fpp2/stationarity.html whereas in this figures (https://i.imgur.com/3lKCxEn.png) the green time series that clearly has cycles (and thus seasonality) is labeled as 'stationary'
(and I have seen these kind of figures quite often if you just google 'stationary time series)

*Can a stationary time series have periods with no fluctuations and periods with high fluctuations? As far as I understood the variance and the (aut)covarianz should not change over time making such a time series not stationary. But here in this picture (https://www.researchgate.net/profile/Hazrat_Ali3/publication/326619835/figure/fig10/AS:654171351044097@1532978012116/Non-stationary-and-stationary-time-series-As-CDR-activities-of-users-are-aggregated-on.png) the time series below is labeld as stationary altough it has periods with changing fluctuations.

I hope you can help me as I am confused about the concept of stationarity. I'd appreciate every comment.
The bounty is to expire quite soon. So I'd be happy if some could at least give me one answer to my questions. It will help me a lot.
Why is nobody answering the questions? Are they not clear enough? If so, please tell me. I think they are important and fundamental as the concept of stationarity is quite important.
 A: *

*Stationary series cannot have a fixed seasonal component, which is saying, if you take a stationary series and you sum it to $sin(t)$, the result will not be stationary. Stationary series can be seasonally autocorrelated, which means that what happens one month is correlated with what will happen the next year in the same month. For stationarity to hold, however, in the long run (after some number of years) this autocorrelation must vanish. It's often hard to tell from sampled data if the time process behind it is stationary or not (it's a matter of statistical tests, not about precise measuring) but that green series in the imgur image does not seem stationary, not anymore than the lower-right red series anyway (the upper two red series show even worse behavior allright).

A premise for the second answer: there is more than one definition of stationarity, but generally both the unconditional mean and variance (and also the auto-covariance function) must be constant over time. This doesn't mean that fluctuations can't happen, but that if you don't know any value of the series around time $t$, knowing $t$ itself doesn't tell you anything about the moments of $Y_t$. This is weak stationarity, strong stationarity is similar but is not limited to first and second order moments (mean, variance, covariance), but the whole distribution. You can relate this to the first answer, as $sin(t)$ would tell you something about the expected value of $Y_t$, so that component makes the series non-stationary.


*In that image both series have strong fluctuations in mean (upper series) or variance (lower series), you may say that the lower series is stationary in mean (that's what Whuber has being saying in the comments), but, as we have seen, this is not sufficient for even the weaker commonly accepted definition of stationarity, that requires also second moment consistency. It's hard to say if either of the two plotted series is stationary, because fluctuations are possible in theory, but must be brief in relation to the length of the whole series, in order to have a good degree of confidence that the series is indeed stationary. In the case of both series plotted in the image that you linked, the "fluctiations" end before the end of the series, and that hints for stationarity, but those could also not be fluctuations at all, but random wandering, they are to long to be considered just fluctuations.

