I have been looking over my notes for Markov chains, and I have come across the following:
Theorem: An irreducible Markov chain is transient iff for some state $i$ there exists a nonzero vector $y$ such that $y_j = \sum_{k\neq i} p_{jk}y_k $, for all $j\neq i$ and $|y_j| \leq 1$for all $j$.
Could you please explain why this is true? I hope to gain some intuition into the result.
Many thanks.