Is this chain irreducible and/or Aperiodic? What is its equilibrium mass function?

Question

Consider a Markov chain with outcomes $\{0,…,n\}$ and transition probabilities

$P_{i,i+1}=p$

$P_{i,i−1}=q$

for $1\le i\le n−1$ and $p+q=1$. Assume also that $P_{0,1} = P_{n,n−1} = 1$. Is this chain irreducible? Aperiodic? Find the equilibrium mass function.

I know if any state can be reached from any other state in a finite # of time steps, a chain is termed irreducible. Also a chain is said to be aperiodic if all states are aperiodic. But I am not sure how to get things going here. Any help would be appreciated.

Answer 1

If $p$ and $q$ are positive, you can conclude that all the states can be reached from all other states. So the chain is irreducible.

However, it is not aperiodic. Because starting from state $i$ you can come back to this state only in even number of steps. So period of state $i$ is 2 for all $i=0, \dots, n$.

For equilibrium mass function, you should solve this matrix equation:

$$ \pi = P \pi , $$ where $\pi$ is the vector of equilibrium probabilities and $P$ is the transition matrix.