Past exam of the probability course of the University of Oxford / 2021 / 3 / b / iii

Suppose a stationary distribution $\pi$ exists. Its stationarity equations areSince ${\sum }_j{\pi }_j=1$, its masses tend to zero as $|j|\to \infty$, and a positive maximum $M={\pi }_{j_0}$ is attained. Both terms on the right are at most $M$. Since $p,q>0$ and $p+q=1$, equality forces ${\pi }_{j_0-a}={\pi }_{j_0+b}=M$. Applying the same argument repeatedly at $j_0-a,j_0-2a,\dots$ givesThese are infinitely many distinct states, contradicting ${\sum }_j{\pi }_j=1$. ThusThis maximum principle proof uses neither a finite expected value of the state nor the coprimality assumption.