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 $X^j=(X_{n,j}{)}_n$ 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 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 $h\ne j$, of the other agents according to the entries of $W$. The best known example of reinforced stochastic process is the Polya urn. The present paper characterizes the asymptotic behavior of the weighted empirical means $N_{n,j}={\sum }_{k=1}^n{q}_{n,k}{X}_{k,j}$, 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 $Z^j=(Z_{n,j}{)}_n$ and for the empirical means ${\overline{X}}^j=({\sum }_{k=1}^n{X}_{k,j}/n{)}_n$ given in recent papers (e.g. [arXiv:1705.02126, Bernoulli, Forth.]; [arXiv:1607.08514, Ann. Appl. Probab., 27(6):3787-3844, 2017]; [arXiv:1602.06217, Stochastic Process. Appl., 129(1):70-101, 2019]). 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 [arXiv:1705.02126, Bernoulli, Forth.]

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