Firstly am I correct in saying that that for an irreducible, aperiodic, positive recurrent Markov chain, a limiting distribution exists, and this distribution is the same as the chain's stationary distribution? (i.e solve $\pi P=\pi$ to find the limiting distribution).
Also, is it true that the limiting and stationary distributions of a Markov chain are always unique? My guess is yes, since we know that solutions to $\pi P=\pi$ are unique up to scalar multiplication, hence if $\pi$ is a distribution then this must mean it is the unique stationary distribution of the chain.