Interacting reinforced stochastic processes: Statistical inference based on the weighted empirical means

Abstract

This work deals with a system of interacting reinforced stochastic processes, where each process X j = (Xn,j )n is located at a vertex j of a finite weighted direct graph, and it can be interpreted as the sequence of "actions" adopted by an agent j of the network. The interaction among the dynamics of these processes depends on the weighted adjacency matrix W associated to the underlying graph: indeed, the probability that an agent j chooses a certain action depends on its personal "inclination" Zn,j and on the inclinations Z n,h , with h = j, of the other agents according to the entries of W . The best known example of reinforced stochastic process is the Pòlya urn.The present paper characterizes the asymptotic behavior of the weighted empirical means Nn,j = n k=1 q n,k X k,j , proving their almost sure synchronization and some central limit theorems in the sense of stable convergence. By means of a more sophisticated decomposition of the considered processes adopted here, these findings complete and improve some asymptotic results for the personal inclinations Z j = (Zn,j )n and for the empirical means X j = ( n k=1 X k,j /n)n given in recent papers (e.g. [1,2,18]). Our work is motivated by the aim to understand how the different rates of convergence of the involved stochastic processes combine and, from an applicative point of view, by the construction of confidence intervals for the common limit inclination of the agents and of a test statistics to make inference on the matrix W , based on the weighted empirical means. In particular, we answer a research question posed in [1].

Vertex reinforced random walks with exponential interaction on complete graphs

“…There is an extensive literature devoted to self-attracting reinforced random walks on graphs, see for instance Pemantle (1992), Benaïm & Tarrès (2011), Volkov (2001), and self-repelling walks, see Toth (1995) and references therein. Several models for interacting generalised Pólya urn models have been considered more recently, see Aletti et al (2020), Crimaldi et al (2019a), Benaïm et al (2015), and van der Hofstad et al (2016). Apart from Chen (2014) and Crimaldi et al (2019b), there are relatively few studies of interacting vertex-reinforced random walks.…”

Interacting non-linear reinforced stochastic processes: synchronization and no-synchronization

“…Note that we exclude the case f LP defined in (4) with θ = 0, because it coincides with the case of a system of interacting Pólya urns with mean-field interaction and with or without a "forcing input" q and this model has been already analyzed in [4][5][6][34][35][36].…”

“…Finally, it is worthwhile to note that the asymptotic behaviour of the stochastic process Z h is strictly related to the one of the stochastic process { Īh n = n k=1 I k,h /n} (see also [5,6]), that is, according to the previous interpretation, the average of times in which agent h adopts the right choice. Therefore, the synchronization or no-synchronization of the inclinations of the agents corresponds to the synchronization or no-synchronization of the average of times in which the agents make the right choice.…”

Interacting Urns on a Finite Directed Graph

“…, f N ) depends on the weighted adjacency matrix of a graph with N nodes, such that each node is thought of as an urn. However, the techniques used in [9] are completely different from the one we used in this paper. We would refer the reader to [11], where the author discusses several techniques used to study problems on random processes with reinforcement and in particular for urn models.…”