Consider an AR(1) model given by
$$X_t=\phi_1X_{t-1}+a_t.$$
AR(1) is stationary tells that all variance and mean of $X$ at any time $t$ should be the same. However, the conditional mean
$$E(X_{t+k}\mid X_t=x_t)=\phi_1^k x_t$$
that indicates the conditional mean is changing with time. Then how this violates the stationarity of $X_t$?
How a stationary random variable becomes not stationary when it comes to conditional?
Thanks for any tips.
The correct definition of stationarity is this:
A process is said to be weakly stationary if both the unconditional mean and the autocovariances do not depend on $t$.
Now, for the first-order autoregression process $X_t = c+\phi X_{t-1}+a_t$, where $a_t$ is white noise with variance $\sigma^2$, the unconditional expected value is
$$\mu_0 = E(X_t) = c/(1-\phi).$$
Furthermore, the unconditional variance is $$ \gamma_0 = \sigma^2/(1-\phi^2), $$ and the $j$th autocovariance is $$ \gamma_j = [\phi^j/(1-\phi^2)]\sigma^2. $$ Thus, since neither the unconditional mean $\mu_0$ nor the autocovariances $\gamma_j$ depend on $t$, we conclude that AR(1) is stationary, provided $|\phi| < 1$.