Past exam of the probability course of the University of Oxford / 2023 / 3 / a / i / Solution

A probability distribution on the countable state space $\mathcal{S}$ is precisely a row vector satisfyingA transition matrix is a stochastic matrix, meaningEvery 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 $P$.