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?