Chapter 9, Problem 9.7PTE

A transition probability matrix is said to be doubly stochastic if ${\displaystyle \underset{i=0}{\overset{M}{\sum }}{P}_{ij}=1}$ for all states $j=0,1,\mathrm{...},M$. Show that such a Markov chain is ergodic, then ${\displaystyle {\prod }_j=}\frac{1}{\left(M+1\right)},j=0,1,\mathrm{...},M.$.