Networks of reinforced stochastic processes: Asymptotics for the empirical means

Abstract

This work deals with systems of interacting reinforced stochastic processes, where each process $X^{j}=(X_{n,j})_{n}$ 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 evolving 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” $Z_{n,j}$ and on the inclinations $Z_{n,h}$, with $h\neq j$, of the other agents according to the elements of $W$.

Asymptotic results for the stochastic processes of the personal inclinations $Z^{j}=(Z_{n,j})_{n}$ have been subject of studies in recent papers (e.g., Aletti, Crimaldi and Ghiglietti [Ann. Appl. Probab. 27 (2017) 3787–3844], Crimaldi et al. [Synchronization and functional central limit theorems for interacting reinforced random walks (2019)]); while the asymptotic behavior of quantities based on the stochastic processes $X^{j}$ of the actions has never been studied yet. In this paper, we fill this gap by characterizing the asymptotic behavior of the empirical means $N_{n,j}=\sum_{k=1}^{n}X_{k,j}/n$, proving their almost sure synchronization and some central limit theorems in the sense of stable convergence. Moreover, we discuss some statistical applications of these convergence results concerning confidence intervals for the random limit toward which all the processes of the system almost surely converge and tools to make inference on the matrix $W$.