Evaluating the long-run component of time-series Consider the following time-series process
$$y_{t+1}=y_{t}-x_t+\epsilon_{t+1},$$
where $\epsilon_{t+1}\sim N(0,\sigma^2)$ (and i.i.d. over time) and $x_t$ is a stationary process. I am interested in forward solutions of $y_t$. Specifically, we can iterate this forward to obtain  
$$y_t=\mathbb{E}_t\sum_{j=0}^{\infty}x_{t+s}+\lim_{s\rightarrow\infty}\mathbb{E}_ty_{t+s},$$
assuming $\mathbb{E}_ty_{t+s}$ is well-defined. How is the term $\lim_{s\rightarrow\infty}\mathbb{E}_ty_{t+s}$ typically evaluated? Also any references to related material are welcome.  
 A: As pointed out in the comments (hat tip to Aksakal and Richard Hardy), if you condition on the sequence $\mathbf{x} \equiv \{ x_t | t \in \mathbb{Z} \}$ then the sequence of interest is a random walk with with this conditioning sequence giving drift terms.  You can write your model in the form of an adjusted random walk as:
$$\Delta y_t = -x_{t-1} + \epsilon_t
\quad \quad \quad \quad \quad 
\epsilon_t \sim \text{IID N}(0, \sigma^2).$$
Now, if $\mathbf{x}$ is even weakly stationary (you stipulate in your question that it is stationary) then you must have a common mean $\mathbb{E}(x_t) = \mu$ so you get $\mathbb{E}(\Delta y_t) = -\mu$, which then gives $\mathbb{E}(y_{s+t}) = \mathbb{E}(y_s) - t \mu$.  The limiting term of interest to you is zero if $\mu=0$ and it diverges if $\mu \neq 0$.

Typically, we would not try to write a random walk in the form you are using, since the limit you are trying to use as a reference point usually diverges.  Instead we would write the value of the time-series relative to a reference point at an arbitrary "starting time".  Using the starting point at time $s$ we get the equation:
$$y_{s+t} 
= y_s + \sum_{i=1}^t \Delta y_{s+i} 
= y_s - \sum_{i=0}^{t-1} x_{s+i} + \sum_{i=1}^t \epsilon_{s+i},$$
which gives the conditional distribution:
$$y_{s+t} | y_s, \mathbf{x}
\sim \text{N} \bigg( y_s - \sum_{i=0}^{t-1} x_{s+i}, \ t \sigma^2 \bigg).$$
If you are willing to stipulate a joint distribution for the values in the sequence $\mathbf{x}$ you can then use the law of total probability to find the conditional distribution of $y_{s+t} | y_s$.
