Hello,
Just two clarifications:
- It is a useful fact that P admits an eigenvalue lambda=-1 if and only if the Markov chain is periodic of period 2. The way to prove this is that the chain has period 2 if and only if the underlying graph is bipartite (i.e. made of two parts A and B, both of which do not have "internal arrows", so that you necessarily jump from A to B or from B to A while making a move in this chain). In this case, you can show that a vector phi=+1 on A and phi=-1 on B is an eigenvector of the transition matrix with eigenvalue -1, and the reciprocal statement ("if there exists an eigenvalue = -1 then the graph is bipartite") also holds.
[Note (5.11.2019): I should add that the above remark holds provided the detailed balance assumption is satisfied]
- Besides, as already said during the lectures, if the matrix P is tridiagonal (i.e. all elements outside then main 3 diagonals are zero), then detailed balance always holds.
Hope this helps,
All the best, and have a nice wek-end,
Olivier
