Quick question on the stationarity of an autoregressive process depending on the time index

by Monolite   Last Updated May 16, 2019 00:19 AM

Let the stochastic process $\{ X_t \}_{t \in \mathbb{Z}}$ satisfy the equation

$$ X_t = \theta X_{t-1} +\epsilon_t $$

where $|\theta| < 1 $ and $\epsilon_t$ is gaussian white noise. This is an AR(1) stationary process.

If we had taken instead of $\{ X_t \}_{t \in \mathbb{Z}}$ with the time index on the integers the time index on the naturals $\{ X_t \}_{t \in \mathbb{N}}$ would we lose the stationarity?

I say this because with the time index on the naturals we obtain $X_t = \sum_{i=0}^t \theta^i \epsilon_{t-i} + \theta^t X_0 $ and the expectation is clearly changing in this case at each time $t$.



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