### 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}A^{k} 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 language | English (US) |
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Article number | 6613535 |

Pages (from-to) | 437-448 |

Number of pages | 12 |

Journal | IEEE Transactions on Automatic Control |

Volume | 59 |

Issue number | 2 |

DOIs | |

State | Published - 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

### Cite this

*IEEE Transactions on Automatic Control*,

*59*(2), 437-448. [6613535]. https://doi.org/10.1109/TAC.2013.2283750