Question

Consider a time-inhomogeneous Markov chain {Xₙ​} n=0,1,2,…​ with state space {0,1,2,3}. Suppose Pr{Xₙ₊₁​=j , Xₙ​=i}={pij​rij​​ when n is even and when n is odd ​ Use a supplementary variable to make this a time-homogeneous chain.

Answer 1

To make a time-inhomogeneous Markov chain time-homogenous, we can introduce a supplementary variable called 'time' and define the transition probabilities as P(i->j) = pij * rij^t, where pij is the transition probability when n is even, rij is the transition probability when n is odd, and t is the time.

Step-by-step explanation:

To make the time-inhomogeneous Markov chain a time-homogeneous chain, we can introduce a supplementary variable called 'time' or 't'. We can then define the transition probabilities as P(i->j) = pij * rij^t, where pij is the transition probability when n is even, rij is the transition probability when n is odd, and t is the time. By introducing this time-modulation factor, we can make the Markov chain time-homogeneous.