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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 finiteweighted direct graph, and it can be interpreted as the sequence of "actions"adopted by an agent $j$ of the network. The interaction among the evolvingdynamics 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 onthe inclinations $Z_{n,h}$, with $h\neq j$, of the other agents according tothe 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 [arXiv:1607.08514, Ann. Appl. Probab.,27(6):3787-3844, 2017]; Crimaldi, Dai Pra, Louis, and Minelli[arXiv:1602.06217, Forthcoming in Stochastic Process. Appl.]); while theasymptotic behavior of quantities based on the stochastic processes $X^j$ ofthe actions has never been studied yet. In this paper, we fill this gap bycharacterizing the asymptotic behavior of the empirical means$N_{n,j}=\sum_{k=1}^n X_{k,j}/n$, proving their almost sure synchronization andsome central limit theorems in the sense of stable convergence. Moreover, wediscuss some statistical applications of these convergence results concerningconfidence intervals for the random limit toward which all the processes of thesystem converge and tools to make inference on the matrix $W$.

Networks of reinforced stochastic processes: asymptotics for the empirical means