A Perron-Frobenius Theorem for Strongly Aperiodic Stochastic Chains

2022/04/01 by Parasnis, Rohit, Franceschetti, Massimo, Touri, Behrouz · 1 citation
#FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2204.00573

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.

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