On indigenous random consensus and averaging dynamics

We study indigenously evolving random averaging dynamics, i.e., random averaging dynamics whose evolution depends on the history of the random dynamics itself. Such dynamical processes find applications in, e.g., models of distributed learning of comparative adjectives in Linguistics, asymmetric state-dependent random gossiping in Computer Science, Hegselmann-Krause opinion dynamics with link-failure and/or random observation radius in Social Sciences, to name just a few. We introduce a novel supermartingale technique to analyze such history-dependent random dynamics. Using this new tool, we show that an adapted random averaging dynamics converges under general conditions and provide a characterization for the asymptotic behavior of such dynamics.