= Solution
A <probability distribution> on the countable <state space> $\mathcal S$ is precisely a row vector satisfying
$$
\boxed{\pi_i\geq0\quad(i\in\mathcal S),\qquad\sum_{i\in\mathcal S}\pi_i=1.}
$$
A <transition matrix> is a <stochastic matrix>, meaning
$$
\boxed{p_{i,j}\geq0\quad(i,j\in\mathcal S),\qquad\sum_{j\in\mathcal S}p_{i,j}=1\quad(i\in\mathcal S).}
$$
Every row therefore gives the conditional <probability distribution> of the next state given the present state. These conditions do not require $\pi$ to be an <invariant distribution> of $P$.
Back to article page
