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I'm just starting to learn the math behind stationarity and differencing, so I apologize if this is a silly question. Lets say I have a non-stationary time series process (pure random walk) defined by:

$Y_t = Y_{t-1} + \varepsilon_{t}$

Where

$\varepsilon_{t} = \text{white noise}$

If I look to make this process stationary through differencing, I subtract $Y_{t-1} from both sides and can write it as:

$Z_t = \varepsilon_{t}$, where $Z_t = Y_t - Y_{t-1}$

If $Z_t$ is now stationary, then I believe the white-noise $\varepsilon_t$ was stationary the whole time.

Is white noise stationary in this context (pure random walk)?

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Yes, white noise is strictly stationary here and in general, and weakly stationary if it has finite second moments (weak stationarity may depend on the precise definition of white noise, i.e. whether the definition assumes finite second moments or not). Wikipedia cites white noise as the simplest example of a stationary process. The thread "Does white noise imply wide-sense stationary?" discusses whether white noise is stationary in more detail.

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