As a property of the markov chain, the stationary distribution has been widely used in many fields like page_rank etc.
However, since the distribution is just a property about the transition matrix and has nothing to do with the init state of the markov chain.
So what's the condition of the transition matrix make the init state has nothing to do with markov chain so it will finally arrive at the stationary distribution after nth iteration.
Markov chains aren't guaranteed to have unique stationary distributions. For example, consider a two state Markov Chain where the transition matrix is the identity matrix. That means that whatever the initial state is, it never changes. So in that case there is no stationary distribution that is independent of the initial case.
Where there is a stationary distribution, unless the initial state is the stationary distribution, the stationary distribution is only reached in the limit as n tends to infinity. So iteration n+1 will be closer to it that iteration n, but however large n is, it won't ever actually be the stationary distribution. However, for practical purposes (i.e. to the limit of the accuracy of floating point numbers in computers), the stationary state may well be reached after a handful of iterations.
You need the underlying graph to be strongly connected and aperiodic. If you want to find the stationary distribution of a periodic Markov chain just by running some chain, add "stay put" transitions with some constant probability to each node and scale the other transitions down appropriately.
