Homogenous Discrete-Time Markov Chain
The homogenity property additionally implies that the state transition probability
pij = P{ Xn= j | Xn-1= i }
will also be independent of n, i.e. the instant when the transition actually occurs.
This implies that the probability of the system going from state i to state j in one step will be the same (i.e. pij ) whenever that happens - this probability will not be different even if it happens at different times.
In this case, the state transition probability will only depend on the value of the initial state and the value of the next state, regardless of when the transition occurs.
Homogenous Discrete-Time Markov Chain
The homogenity property also implies that we can define a m-step state transition probability as follows -
| pij(m) = P{ Xn+m = j | Xn =i}= |  |
pik(m-1)pkj | m=2, 3,........... |
where pij(m) is the probability that the state changes from state i to state j in m steps. The way the above equation is written, we go from state i to k in (m-1) steps and then go from k to j in the last (mth) step, and sum the probabilities of each of these over all possible intermediate states (i.e. k).
This may also be written in other ways, though the value obtained in each case will be the same.
For example, we can also write pij(m) in the following way.
| pij(m) = P{ Xn+m = j | Xn =i}= | |
pik pkj(m-1) m=2, 3,......... |
This way of expressing the m-step state transition probability is one where the system goes from state i to state k in the first step and then goes from state k to state j in the next (m-1) steps. The probabilities of each of these are summed over all possible intermediate states.