I have just learnt about the invariant distribution, and was wondering if such depends on the initial distribution of a Markov chain in anyway, and if so, whether the following is the correct reasoning:
The invariant distribution is a distribution $\pi$ such that $\pi P=\pi$, where $P$ is the transition probability matrix. The transition probability matrix doesn't depend on the initial distribution, according to my understanding; therefore the invariant distribution doesn't depend on the initial distribution as it depends only on the probability transition matrix (we could determine the invariant distribution just given the probability transition matrix).
Is this correct?
If $\ P\ $ is the transition matrix of a finite-state homogeneous Markov chain then any convex combination of the rows of the limit $$ Q=\lim_\limits{n\rightarrow\infty}\frac{1}{n}\sum_{j=1}^n P^j $$ (which always exists) will be an invariant distribution of the Markov chain. However, the rows of $\ Q\ $ are all the same if and only if $\ P\ $ is irreducible, and the limit $\ \lim_\limits{n\rightarrow\infty}P^n\ $ (which must equal $\ Q\ $ whenever it exists) exists if and only if $\ P\ $ is aperiodic. It follows from this that a finite-state homogeneous Markov chain has a unique invariant distribution, regardless of its initial distribution, if and only if its transition matrix is irreducible and aperiodic.
When $\ P\ $ is irreducible but periodic, there will exist initial distributions $\ \pi\ $ for which the limit $\ \lim_\limits{n\rightarrow\infty}\pi P^n\ $ doesn't exist. In such cases, the state distribution of the chain doesn't converge to an invariant distribution. However, there will also always exist initial distributions $\ \pi\ $ for which the sequence $\ \pi P^n\ $ does converge as $\ n\rightarrow\infty\ .$ Its limit must be then the common value of the rows of $\ Q\ ,$ independent of $\ \pi\ $ (as long as the limit does exist).
When $\ P\ $ is reducible, the sequence $\ \pi P^n\ $ might converge or it might not. When it does converge the limiting distribution will be an invariant distribution of the chain, but its value will depend on the particular initial distribution chosen.