  The Perron-Frobenius theorem plays an important role in many areas of
management science and operations research. This paper 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 provides conditions in both the
finite-dimensional and infinite-dimensional settings under which the
Perron-Frobenius eigenvalue and eigenvectors exist. Furthermore, 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.
