Is it possible for a Markov chain to have 2 or more invariant distributions? I'm trying to understand the Perron-Frobenius Theorem, and it seems like it's saying that an eigenvector of P is unique as long as its components are positive? Of course, there can be infinitely many of these distributions too, but I am concerned with the finite case. I would love some insight on why it is not possible to have multiple invariant distributions for Markov chains!
2026-04-04 01:48:21.1775267301
Markov Chains with multiple invariant distributions
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The set of stationary distributions is a convex set. If it is not a singleton, it contains the line segment connecting any two distinct stationary distributions. So its cardinality cannot be a finite number greater than 1. Either it's unique or there are infinitely many of them.