### 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) |
---|---|

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 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

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

### Cite this

*IEEE Transactions on Automatic Control*,

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

**Product of random stochastic matrices.** / Touri, Behrouz; Nedich, Angelia.

Research output: Contribution to journal › Article

*IEEE Transactions on Automatic Control*, vol. 59, no. 2, 6613535, pp. 437-448. https://doi.org/10.1109/TAC.2013.2283750

}

TY - JOUR

T1 - Product of random stochastic matrices

AU - Touri, Behrouz

AU - Nedich, Angelia

PY - 2014/2

Y1 - 2014/2

N2 - 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.

AB - 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.

KW - Balanced

KW - consensus

KW - product of stochastic matrices

KW - random connectivity

KW - random matrix

UR - http://www.scopus.com/inward/record.url?scp=84893596333&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84893596333&partnerID=8YFLogxK

U2 - 10.1109/TAC.2013.2283750

DO - 10.1109/TAC.2013.2283750

M3 - Article

AN - SCOPUS:84893596333

VL - 59

SP - 437

EP - 448

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

SN - 0018-9286

IS - 2

M1 - 6613535

ER -