We provide new insight into the somewhat obscure definition of the Sarymsakov class of stochastic matrices and use it to construct a new necessary and sufficient condition for the convergence of products of stochastic matrices. Such convergence result is critical in establishing the effectiveness of distributed coordination algorithms for multi-agent systems and enables us to investigate a specific coordination task with asynchronous update events. The set of scrambling stochastic matrices, a subclass of the Sarymsakov class, is utilized to establish the convergence of the agents' states even when there is no common clock for the agents to synchronize their update actions.