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Consider the following AR(1) process with i.i.d. normal errors that have zero mean and finite variance $\sigma^2>0$, $$x_t = (1-\rho)\mu + \rho x_{t-1} + \epsilon _t$$ Now suppose $\rho = 1/2$ and $\mu = 1$. This process does not have a unit root, and it is not mean stationary. At any point in time, the process has finite variance, although as time ...
Write the series in the answer as $(x_t - \mu) = \rho (x_{t-1} - \mu) + \varepsilon_t$ then if $\rho=.5$ and $\varepsilon_t$ is $N(0,\sigma^2)$, $(x_t - \mu)$ is stationary with mean $0$ and variance $\frac{\sigma^2}{1-\rho}$. A time series process can have a deterministic part and a pure random part. The definition of stationarity (strict or strong or ...