I have modeled a process with a Markov chain with K+1 states which is irreducible and apperiodic. The transition matrix is a centrosymmetric matrix where all it's entries has a positive probability. I'm trying to prove that the stationary distribution has the shape of a "Bell" (After plotting the discrete distribution in Mathematica), which means the distribution has a maximum at the middle state and from state 0 to K/2 it is monotonically increasing and form K/2 and on it is monotonically decreasing. I'v proven that the stationary distribution is symmetric around the middle state.
From the transition matrix it turns out that - given I'm in a specific state the distribution for the next state is "Bell" shaped with its maximum not necessarily in the center. The question is if for every state I have the same behavior for the next state distribution, does the stationary distribution necessarily act the same?
For example for a n*n transition matrix where all of its entries are 1/n the next state distribution is uniform for all states and so is the stationary distribution.
Thanks