I have a question regarding ergodicity in the context of Markov chains. I do understand that the intuition about it is that if it’s possible to get from any state to any other state you’ve got an ergodic chain, fine.
My lecturer (who uses Ljungqvist and Sargent’s ‘recursive macroeconomic theory’) handed this algorithm to verify whether a MC is ergodic down to us:
a) find the stationary distribution b) find the invariant function(s) c) check that all the invariant function(s) are constant under the stationary distribution -if so, you can say that the MC is ergodic.
I have a problem with c) – this is probably due to the fact that I’m not a mathematician, but I don’t understand how one goes about that.
He supplied an example- if someone can explain to me, as if I were a child (which, mathematically, I am), how c) is fulfilled there, I would be very grateful.
The transition matrix is (0.8 0.2 0; 0.1 0.9 0; 0 0 1)
It’s straightforward to verify that the stationary distribution, call it π’, is: [1/3γ(1), 2/3γ(2), 1-γ] for any γ∈ [0,1]. It’s equally simple to work out that the invariant functions work out to be: y(1) = y(2), y(3) any
I’m then told that if γ=0, γ=1, that the Markov chain is ergodic, whilst for γ∈ (0,1), the Markov chain is not ergodic. How does that work?