How to interpret hint in Q3 of the additional questions

48 views

I have managed to prove that this is true by induction for both cases γ12 + γ21 = 1 and γ12 + γ21 = 2, but don't understand why we only have to consider these two cases? I can see that if γ12 + γ21 = 1 then it is periodic, struggling to interpret how γ12 + γ21 = 2 implies independence?

answered May 14 by (610 points)
selected May 14

The probability of moving from one state to another lies on the interval [0 ; 1].

The only way for r12 + r21 = 2, is if r12 = r21 = 1.

(Draw this markov chain to see that there is only one possible transition path from each state.)

This implies independence since the probability of moving to state 1 from state 2, is independent of the transition history. It does not matter on the earlier states since there is only possible transition path from state 2 to state 1 - i.e. this transition will happen with probability 1.

Likewise for moving to state 2 from state 1.