Let $\Pi$ be a stochastic matrix of an irreducible markov chain.
We define the spectral radius of $\Pi$ as:
$\rho(\Pi) := \limsup_{n \to \infty} \left( \pi^{(n)}_{(a,b)} \right)^{\frac{1}{n}}$
Why is this definition independent of the choice of states $a$ and $b$?
Or is it, that the definition is imprecise and should be $\lim_{n\to\infty}\sup_{(a,b)}$?
Am I right, that $\rho(\Pi) \in [0,1]$?