2011/10/08 by Behrouz Touri, Touri, Behrouz, Angelia Nedich +1 · 1 citation
Computer Science · Mathematics · #Neural Networks Stability and Synchronization #Topological and Geometric Data Analysis #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1110.1751
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 \Pstar, which plays a central role in this work. We then introduce balanced chains and using some geometric properties of these chains, we characterize the stability of a subclass of 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 limk→∞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.