Is this AR(1) series stationary?

Question

A series follows Xt = 0.8\,X{t-1} + \varepsilont, where \varepsilont is white noise with mean 0 and variance \sigma^2, and the process has been running from the distant past. Is \{Xt\} weakly stationary? Answer yes or no, and justify.

Accepted Answer

For an AR(1) process Xt = \phi X{t-1} + \varepsilont, a stationary solution exists exactly when \phi < 1. Here \phi = 0.8, so the condition holds and each shock's influence fades geometrically instead of piling up. Under \phi < 1 the stationary statistics are: - Mean: E[Xt] = 0, constant. - Variance: \operatorname{Var}(Xt) = \dfrac{\sigma^2}{1-\phi^2} = \dfrac{\sigma^2}{1-0.64} = \dfrac{\sigma^2}{0.36}, finite and constant. - Autocorrelation: \rhok = \phi^k = 0.8^k, which depends only on the lag k. All three stationarity conditions are met, so the series is weakly stationary. Contrast this wi…

Is this AR(1) series stationary?

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