How does one show that any finite-state time homogenous Markov Chain has at least one stationary distribution in the sense of $\pi = \pi Q$ where $Q$ is the transition matrix and $\pi$ is the stationary distribution? My instinct is it involved eigenvalues and the Perron Frobenius theorem but I'm having trouble completing the argument.
There are a number of ways to prove this. In a recent article "What is a stationary measure?", Alex Furman outlines a straightforward proof using only linear algebra. He notes that the existence of an invariant probability measure also follows from the Brouwer fixed point theorem.
Isn't it simple? Suppose you take the matrix to a power k where k-> infinity. Then all eigenvalues approach 0 while eigenvalue =1 is stationary. Note that the image is always the span of colloms so for eigenvalue 1 the picture must be one dimensional.
Any stochastic matrix $P$ has at least one stationary distribution $\pi$ in the sense that $\pi P=\pi$. This can be shown by the following statement from Wikipedia.
Brouwer Fixed Point Theorem (applied to the compact convex set of all probability distributions of the finite state space $\Omega$) implies that there is some left eigenvector which is also a stationary probability vector.
Of course, we could also prove this by Perron Frobenius theorem.