Suppose we are given $x_t, \bar{x_t}, t\in \mathbb{Z_+}$ independent 2-states $\{0, 1\}$ Markov chains with positive transition probabilities. Initial states are $x_0 = 0; \bar{x}_0 = 1$. For which positive real numbers $a, b$ random process $\eta_t = ax_t + b\bar{x_t}$ is stationary?

I know that stationary distribution for one particular Markov chain would be $\pi = (0.5, 0.5)$. How to proceed from here? Or instead i need to check that distribution of $\eta _t$ is translation invariant?



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