Problem: Given the Ehrenfest chain $(X_n)_n$ with the state space $S=\left\{0,1,\ldots,m\right\}$. For the basic Ehrenfest chain, the limiting behavior of the chain is as follows:
- $P^{2n}(x,y) \to C^k_m \left(\dfrac{1}{2}\right)^{m-1}$ as $n\to \infty$ if $x,y \in S$ have the same parity (both even or both odd). The limit is zero otherwise.
- $P^{2n+1}(x,y) \to C^k_m \left(\dfrac{1}{2}\right)^{m-1}$ as $n \to \infty$ if $x,y \in S$ have opposite parity (one even and one odd). The limit is zero otherwise.
My attempt: I have found that the limit seem similar to the stationary distribution formula which is $$\nu_k = C^k_m \left(\dfrac{1}{2}\right)^{m}.$$
But I have not think that there is any relation between them.