In this chapter, we discuss some of the implications of the results developed in Chaps. 3 and 4. We first study the implications of the developed results on the product of independent random stochastic chains in Sect. 5.1, and also develop a rate of convergence result for ergodic independent random chains. Then in Sect. 5.2, we study the implication of Theorem 4.10 in non-negative matrix theory. There, we show that Theorem 4.10 is an extension of a well-known result for homogeneous Markov chains to inhomogeneous products of stochastic matrices. In Sect. 5.3, we provide a convergence rate analysis for averaging dynamics driven by uniformly bounded chains. Then, in Sect. 5.4, we introduce link-failure models for random chains and analyze the effect of link-failure on the limiting behavior of averaging dynamics. In Sect. 5.5, we study the Hegselmann-Krause model for opinion dynamics in social networks and provide a new bound on the termination time of such dynamics. Finally, in Sect. 5.6, using the developed tools, we propose an alternative proof for the second Borel-Cantelli lemma.
KeywordsConvergence Result Termination Time Final Component Trivial Event Stochastic Matrice
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