Testing for stationarity in time series
One of the most important concepts in time series analysis is stationarity. Plainly speaking a stationary time series is a series whose properties do not depend on the time at which the series is observed. In other words, stationarity implies that the statistical properties of the data-generating process (DGP) of a certain time series do not change over time.
Hence, we should not be able to see any trend or seasonal patterns in a stationary time series, as their existence violates the stationarity assumptions. On the other hand, a white noise process is stationary, as it does not matter when we observe it, it will always look pretty much the same at any point in time.
A time series without trend and seasonality but with cyclic behavior can still be stationary because the cycles are not of a fixed length. So unless we explicitly observe a time series, we cannot be sure where the peaks and troughs of the cycles will be located.
To put it more formally...
