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I was just wandering if, given the AR process \begin{equation} X_t = \alpha X_{t-1} + \varepsilon_t, \quad \varepsilon_t \overset{iid}{\sim} N(0,1), \end{equation} the $X_t$ values are independent due to the independent white noise?

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They are dependent. One way to see that is that the correlation between $X_t$ and $X_{t-1}$ is a constant that is equal to zero only if $\alpha = 0$. Another way to put it is that the conditional distribution of $X_t$ given $X_{t-1}$ it dependent on $X_{t-1}$ simply because has a mean equal to $\alpha X_{t-1}$.

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