Let n > 0 and Xn be an irreducible aperiodic Markov chain having a doubly stochastic transition matrix.
By definition, $\sum_{y∈S} P(x,y) = 1$ and $\sum_{x∈S} P(x,y) = 1$ for all x y ∈ S.
I want to show that if |S| < $\infty$ then $P^{n}(x,y)\rightarrow 1/|S|$ as $n\rightarrow \infty$
Could somebody please help? Thanks in advance!!!
Best Answer
A basic theorem in MC theory shows that the limit exists and is independent of $x$. Let $\pi (y)=\lim P^{n}(x,y)$. Now $\sum_{x \in S} P(x,y)=1$ also gives $\sum_{x \in S} P^{n}(x,y)=1$ for all $n$. Since this is a finite sum we can take limit as $n \to \infty$ to see that $\sum_{x \in S} \pi (y)=1$ which means $|S| \pi (y)=1$ or $\pi (y)=\frac 1 {|S|}$.
Reference for the limit theorem: Theorem III.2.1, p. 67 of 'Markov Chains: Theory and Applications' by Isaakson and Madsen.