Product of random stochastic matrices

Behrouz Touri, Angelia Nedic

Research output: Contribution to journalArticle

57 Scopus citations

Abstract

The paper deals with the convergence properties of the products of random (row-)stochastic matrices. The limiting behavior of such products is studied from a dynamical system point of view. In particular, by appropriately defining a dynamic associated with a given sequence of random (row-)stochastic matrices, we prove that the dynamics admits a class of time-varying Lyapunov functions, including a quadratic one. Then, we discuss a special class of stochastic matrices, a class ${\cal P}\ast, which plays a central role in this work. We then study cut-balanced chains and using some geometric properties of these chains, we characterize the stability of a subclass of cut-balanced chains. As a special consequence of this stability result, we obtain an extension of a central result in the non-negative matrix theory stating that, for any aperiodic and irreducible row-stochastic matrix $A$ , the limit $\lim-{k\rightarrow\infty}Ak exists and it is a rank one stochastic matrix. We show that a generalization of this result holds not only for sequences of stochastic matrices but also for independent random sequences of such matrices.

Original languageEnglish (US)
Article number6613535
Pages (from-to)437-448
Number of pages12
JournalIEEE Transactions on Automatic Control
Volume59
Issue number2
DOIs
StatePublished - Feb 2014

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Keywords

  • Balanced
  • consensus
  • product of stochastic matrices
  • random connectivity
  • random matrix

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Computer Science Applications
  • Electrical and Electronic Engineering

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