# When does $\lim_{j\to \infty} \mathbb{E}_t[X_{t+j}]$ exist?

Consider a univariate stochastic process (time series) $$X_t$$. I am interested in conditions under which $$\lim_{j\to \infty} \mathbb{E}_t[X_{t+j}]$$ exists. For example if $$X_t$$ is a stationary process it seems that $$\lim_{j\to \infty} \mathbb{E}_t[X_{t+j}]$$ is constant. Is this correct? Are there other interesting cases? I have been told that the existense of this limit might be related to "conditional stationarity".

Intuitively $$\mathbb{E}_t[X_{t+j}]$$ should approach its long run unconditional mean (which is constant due to stationarity). However, I am not sure how to prove this result. An example of a strictly stationary process for which this property is true would be the standard AR(1) process $$X_t=\rho X_{t-1}+\epsilon_t$$, where $$-1<\rho<1$$ and $$\epsilon_t$$ is Gaussian white noise.

Another interesting case? If $$X_t$$ is a martingale, then for any $$j$$ $$E[X_{t+j}|X_t]=X_t$$ and so $$E[X_{t+j}]=E[X_t]$$ but you typically don't have stationarity (even approximately)