This paper presents new sufficient conditions for convergence and asymptotic or exponential stability of a stochastic discrete-time system, under which the constructed Lyapunov function always decreases in expectation along the system's solutions after a finite number of steps, but without necessarily strict decrease at every step, in contrast to the classical stochastic Lyapunov theory. As the first application of this new Lyapunov criterion, we look at the product of any random sequence of stochastic matrices, including those with zero diagonal entries, and obtain sufficient conditions to ensure the product almost surely converges to a matrix with identical rows; we also show that the rate of convergence can be exponential under additional conditions. As the second application, we study a distributed network algorithm for solving linear algebraic equations. We relax existing conditions on the network structures, while still guaranteeing the equations are solved asymptotically.

中文翻译：

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随机系统的李雅普诺夫判据及其在分布式计算中的应用

本文提出了随机离散时间系统收敛和渐近或指数稳定性的新的充分条件，在该条件下，构造的李雅普诺夫函数总是在有限步数后沿系统解的期望值递减，但不一定在每一步都严格递减。 ，与经典随机李雅普诺夫理论相反。作为这个新李雅普诺夫准则的第一个应用，我们查看随机矩阵的任何随机序列的乘积，包括那些对角项为零的随机矩阵，并获得充分条件以确保乘积几乎肯定收敛到具有相同行的矩阵；我们还表明，在附加条件下收敛速度可以是指数级的。作为第二个应用程序，我们研究了一种用于求解线性代数方程的分布式网络算法。我们放宽了网络结构上的现有条件，同时仍然保证渐近求解方程。