Product of Random Stochastic Matrices
Product of Random Stochastic Matrices
复制标题
DOI：
10.1109/tac.2013.2283750
复制
发表时间：
2011-10
影响因子：
6.8
通讯作者：
B. Touri;A. Nedić
中科院分区：
文献类型：
--
作者：
B. Touri;A. Nedić
本文研究了随机（行）随机矩阵乘积的收敛性质。从动力学系统的角度研究了这类产品的极限行为。特别是，通过适当地定义一个动态与一个给定的序列的随机（行）随机矩阵，我们证明了动态承认一类时变李雅普诺夫函数，包括二次的。然后，我们讨论了一类特殊的随机矩阵，一类P*，这在这项工作中起着核心作用。然后，我们研究割平衡链，并利用这些链的一些几何性质，我们刻画了一类割平衡链的稳定性。作为这个稳定性结果的一个特殊结果，我们得到了非负矩阵理论中一个中心结果的推广，即对任意非周期不可约行随机矩阵A，极限limk→∞ Ak存在，并且它是一个秩为1的随机矩阵.我们证明了这一结果的推广不仅适用于随机矩阵序列，而且适用于独立的随机序列。
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 P*, 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 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.