Limit Behavior of One Random Medium

Each lamp in a row infinite in both directions continues to burn at the next instant if at the given instant it, together with a neighboring lamp, is on; the lamp goes on with probability $\theta$ in other cases ($\theta$ is a small positive number). It is shown that for any initial state of this medium, the probability distribution on the state space asymptotically approaches a linear combination of two fixed stationary distributions, one corresponding to the “all lamps on” state, while the other is regular.