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

Question

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.

Answer 1

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)