A Perron-Frobenius Theorem for Strongly Aperiodic Stochastic Chains
Abstract
We derive a generalization of the Perron-Frobenius theorem to time-varying row-stochastic matrices as follows: using Kolmogorov's concept of absolute probability sequences, which are time-varying analogs of principal eigenvectors, we identify a set of connectivity conditions that generalize the notion of irreducibility (strong connectivity) to time-varying matrices (networks), and we show that under these conditions, the absolute probability sequence associated with a given matrix sequence is (a) uniformly positive and (b) unique. Our results apply to both discrete-time and continuous-time settings. We then discuss a few applications of our main results to non-Bayesian learning, distributed optimization, opinion dynamics, and averaging dynamics over random networks.
Cite
@article{arxiv.2204.00573,
title = {A Perron-Frobenius Theorem for Strongly Aperiodic Stochastic Chains},
author = {Rohit Parasnis and Massimo Franceschetti and Behrouz Touri},
journal= {arXiv preprint arXiv:2204.00573},
year = {2024}
}