Eigenvalues and Nonhomogeneous Markov Chains
When the state space is finite, we can rely on the standard results of linear algebra to study the asymptotic behavior of homogeneous Markov chains. Indeed, the asymptotic behavior of the distribution at time n of the chain is entirely described by the asymptotic behavior of the n-step transition matrix P n and the latter depends on the eigenstructure of P. The Perron-Frobenius theorem detailing the eigenstructure of nonnegative matrices is therefore all that is needed, at least in the theory.
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