A probabilistic proof of the Perron-Frobenius theorem

The Perron-Frobenius theorem plays an important role in many areas of management science and operations research. This article provides a probabilistic perspective on the theorem, by discussing a proof that exploits a probabilistic representation of the Perron-Frobenius eigenvalue and eigenvectors in terms of the dynamics of a Markov chain. The proof recovers conditions in both the finite-dimensional and infinite-dimensional settings under which the Perron-Frobenius eigenvalue and eigenvectors have been shown to exist by other methods. In addition to providing new insights, the probabilistic representations that arise can be used to produce a Monte-Carlo algorithm for computing the Perron-Frobenius eigenvalue and eigenvectors that will be explored elsewhere.