Fractional Gaussian noise (fGn) is characterized by the mean ($\mu$), the standard deviation ($\sigma$), and the Hurst index ($H$). It's my understanding that it is stationary, for the simple reason that the mean, variance, and correlation structure are constant over time.
I also understand that the KPSS test (Kwiatkowski et al., 1992) can be used to check if a time series is stationary. However, the null hypothesis (which, if accepted, corresponds to a decision of 'stationary') is that $\sigma^2 = 0$, versus the alternative that $\sigma^2 > 0$.
In my view, this leads to a contradiction. On one hand, the fGn time series is considered stationary because it has a constant variance, etc. On the other hand, the KPSS test considers a time series to be stationary only if no significant variance is present. Are two different meanings of stationarity involved here? Or have I misunderstood something else?