Why does markov chains power method converge at the rate of |λ_2/λ_1 |

Question

I'm doing some researches on Markov Chains, and every time I meet this statement, that The rate of convergence of the power method is given by |λ_2/λ_1 |^k→0, when k→inf. And where λ_1 and λ_2 are the biggest eigenvalues of the irreducible transition matrix

Can someone please explain to me how they landed to this.

Answer 1

For you intuition:

Imagine that the initial state is a (weighted) sum, a mixture, of some or all of the eigenstates. Each eigenvalue represents the "rate of decay" of the presense of each eigenstate in the "current state" of the chain as the chain evolves. When all states but the first are negligible you are left practically with the first eigenstate which is the equilibrium state. The ratio $\lambda_2/\lambda_1$ expresses how quickly that happens.

For a formal derivation (and much more stuff to study) a standard textbook is Norris' "Markov chains".