# Product of random stochastic matrices

Behrouz Touri, Angelia Nedich

Research output: Contribution to journalArticle

55 Citations (Scopus)

### 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 language English (US) 6613535 437-448 12 IEEE Transactions on Automatic Control 59 2 https://doi.org/10.1109/TAC.2013.2283750 Published - Feb 2014 Yes

### Fingerprint

Lyapunov functions
Dynamical systems

### 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

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

In: IEEE Transactions on Automatic Control, Vol. 59, No. 2, 6613535, 02.2014, p. 437-448.

Research output: Contribution to journalArticle

@article{9f8f6d87e2914bc38967c2febc6ad69b,
title = "Product of random stochastic matrices",
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.",
keywords = "Balanced, consensus, product of stochastic matrices, random connectivity, random matrix",
author = "Behrouz Touri and Angelia Nedich",
year = "2014",
month = "2",
doi = "10.1109/TAC.2013.2283750",
language = "English (US)",
volume = "59",
pages = "437--448",
journal = "IEEE Transactions on Automatic Control",
issn = "0018-9286",
publisher = "Institute of Electrical and Electronics Engineers Inc.",
number = "2",

}

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 -