A few days ago, I saw a post showing how the stationary distribution of an infinite-state Markov chain can be computed. Since for a finite-state Markov chain, we can solve its eigenvalue equation to get the stationary distribution (which is an eigenvector), I'm wondering if we can also solve for eigenvalues/eigenvectors for an infinite-state Markov chain. If so, how is it different from solving the eigenvalue equation from a finite-state case? If not, is there a way that we could assess the mixing time for an infinite-state Markov chain? (For a finite-state case, we can just calculate the spectral gap.)
2026-03-25 23:34:52.1774481692
Do infinite state Markov chains have eigenvalues?
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You can't really do much with an infinite-state Markov chain in general, but in some cases you can express the eigenvalue equation as a recurrence and solve that.