Time reversible Markov Chain

A total of m white and m black balls are distributed among two urns, with each urn containing m balls. At each stage, a ball is randomly selected from each urn and

the two selected balls are interchanged. Let denote the number of black balls in urn 1 after the th interchange.

(a) Give the transition probabilities of the Markov chain , .

(b) Without any computations, what do you think are the limiting probabilities

of this chain?

(c) Find the limiting probabilities and show that the stationary chain is time

reversible.

Attempted solution:

Transition probabilities

I think those are the transition probabilities. Where is the current number of black balls.

I also know that in order to be time reversible I need the stationary distribution to satisfy: , where is the stationary distribution.

Any assistance as to how I can guess the correct form of the solution based on intuition regarding this scenario would be helpful, thanks in advance.