A probabilistic proof of the Perron–Frobenius theorem

Abstract

AbstractThe 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.