A probabilistic proof of Perron's theorem

TL;DR

This paper offers a probabilistic proof of Perron's theorem by representing the Perron eigenvector through the trajectories of an auxiliary Markov chain, providing a novel perspective on the classical result.

Contribution

It introduces a new probabilistic approach to prove Perron's theorem using Markov chain trajectories, differing from traditional algebraic proofs.

Findings

Eigenvector represented as a functional of Markov chain trajectories

Provides an alternative probabilistic proof of Perron's theorem

Highlights the connection between eigenvectors and stochastic processes

Abstract

We present an alternative proof of Perron's theorem, which is probabilistic in nature. It rests on the representation of the Perron eigenvector as a functional of the trajectory of an auxiliary Markov chain.