Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 1 / 19H / c

For the converse as well as the stated symmetric case, setIrreducibility and the existence of an invariant probability distribution make the chain positive recurrent, so $a,b>0$. During one return cycle from $i$ to $i$, the chain enters $j$ with probability $a$ and, after entry, makes a geometric number of visits to $j$ with success parameter $b$. ThereforeOn the other hand, the stationary cycle occupation formula givesConsequentlyIt follows thatwhich is precisely the claimed equivalence between symmetry of $i,j$ and equality of their invariant masses.