Ok so we are given a Markov chain $X_n$, $P=P(ij)$ as the transition matrix and the $(\pi_1,\pi_2,\pi_3,…,\pi_n)$ as steady-state distribution of the chain. We are asked to prove that for every $i$:
$\sum_{i\neq j} \pi_iPij = \sum_{j\neq i} \pi_jPji$
Can somebody explain me what that means and why it is (obviously true for every steady -state distribution? Plus, how do we prove it??
Best Answer
I think you need the condition that the Markov Chain is reversible.
Suppose that $\mathbb{\pi}$ satisfies these conditions, then
$$\sum_{i} \pi_iPij = \sum_{i} \pi_jPji = \pi_j \sum_{i} Pji = \pi_j $$