This question relates to a theorem about the decomposition of additive functionals---a technique especially useful in macroeconomics and finance. This question has two objective. First, I don't have a reference to the theorem that shows when this is possible, and I would like to find it. Second, I would like to know how to actually find this decomposition.
Consider the following autoregressive model: $$ X_{t+1} = \alpha_0 + \beta_0 (X_t - \alpha_0) + W_{t+1}, $$ where $-1 < \beta_0 < 1$ and $W_{t+1}$ is distributed as a normal with mean zero and variance one. Now, consider the additive functional $$ Y_t = \sum_{j=1}^t X_j. $$ How would I produce the decomposition of the form $$ Y_t = r_1 t + M_t - r_2 (X_t - X_0) $$ where $\{M_t \}$ is a Martingale? (How can obtain the values $r_1$, $r_2$, and $M_t - M_{t-1}$?)