I am currently relooking at some basics of Markov Chain (after a long time) and find myself confused over some concepts.
Defintion: stationary distribution
Let $P$ be a distribution of some dimension $R^{d}$ and $T$ a transition probability matrix. A distribution $P$ is a stationary distribution $P_{s}$ if the eigenvector equation $P^{m}T = P^{m}$ is satisfied, where $m$ denotes the iteration index. If the eigenvector equation holds true, then my probability distribution is a stationary distribution, isn't it?
How is the above definition different to the theorem for the existence of a stationary distribution? Is there a connection?
The standard definition of a stationary distribution $P$ of a transition matrix $T$ is its "normalized" positive (non-negative) eigenvector(s) associated with the eigenvalue $1$, i.e. \begin{align} PT = P. \end{align} This always exists.
Although less common (at least for me), is the definition you have given: a distribution is said to be stationary (more precisely, eventually stationary, not a standard term) if there exists $m\in \mathbb{N}$ such that \begin{align} P^m T = P^m \end{align} where $P^m := P T^{m-1}$. Of course, the set of eventually stationary distribution encompasses the set of stationary distribution, i.e. $m=1$ case.
Notice the definition of eventually stationary distribution requires the existence of a finite $m$ such that $P^m T = P^m$. It is not clear if you can even find a $P$ such that $P^2$ is an eigenvector with eigenvalue $1$ for $T$ (of course, we assume $P$ is not an eigenvector), or more generally, it is not clear how one can find $P$ such that $P^m$ is an eigenvector. But even so, this definition eventually stationary distribution is clearly more general than the first definition.