A Perron-Frobenius Theorem for Strongly Aperiodic Stochastic Chains

• 成果类型: Article
• 署名作者: Parasnis, Rohit; Franceschetti, Massimo; Touri, Behrouz
• 署名单位: University of California System; University of California San Diego
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2025.3527332
• 发表日期: 2025
• 页码: 4286-4301
• 关键词: Time-varying systems; vectors; Heuristic algorithms; Stochastic processes; Sufficient conditions; Linear matrix inequalities; dynamical systems; CONVERGENCE; optimization; indexes; Consensus algorithms; decentralized control; Distributed algorithms; networked control systems; Time-varying systems

摘要

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.