Question

What happens if a markov chain doesn't starts with stationary distribution

Answer 1

I'm currently trying to understand (intuitively) what a stationary distribution of a Markov Chain is? In our lecture notes, we're given the following definition:

def

This was of little benefit to my understanding, so I've tried searching online for a more useful explanation. I then found the following video, which improved my understanding to the extent that I now understand that stationary distributions are to do with looking at what happens to the probabilities at each state within a Markov Chain when time becomes infinitely large. This is still not a sufficient enough understanding of the concept though.

For example, I've been asked to show that

πa=(25,35,0,0,0)πb=(0,0,1,0,0)πc=(0,0,0,35,25)

are stationary distributions with respect to the Markov Chain with one-step transition martix

P=⎛⎝⎜⎜⎜⎜⎜⎜⎜⎜121300012230000010000023120001312⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟

How would you do this? What is a stationary distribution, with respect to this example?

Also, could someone please confirm that I'm correct in thinking that the notation pij denotes the probability of the process moving from the state i to the state j?