I'm a little unsure if my approach here is correct, which is making it difficult to find the limit. Any and all help will be appreciated!
Let be a primitive finite state Markov chain with transition matrix and stationary distribution . Define the process by:
Show that is a Markov chain, and compute:
What I've done:
So I need to show that is only determined by , ie:
So, suppose a & b are just some numbers. Then:
Plugging in the definition of :
=
Because are primitive Markov-Chains, can I say that
=
because doesn't depend on ? This is basically where I'm stuck