A probabilistic proof of the Perron–Frobenius theorem
本文用马尔可夫链的概率表示来证明Perron-Frobenius定理，适用于有限维和无限维情形，并可用于设计蒙特卡洛算法计算特征值和特征向量。
Abstract 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.