# Lyapunov Criterion for Stochastic Systems and Its Applications in Distributed Computation

## TL;DR

This paper introduces a new Lyapunov criterion for stochastic systems that guarantees convergence and stability, with applications in analyzing random matrix products and distributed algorithms for solving linear equations.

## Contribution

It proposes a novel Lyapunov condition allowing finite-step expected decrease without strict per-step decrease, extending classical stochastic stability theory.

## Key findings

- Conditions for almost sure convergence of random matrix products.
- Exponential convergence rate under additional assumptions.
- Relaxed network structure requirements for distributed linear algebra algorithms.

## Abstract

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.