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

by   Giacomo Aletti, et al.

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≠ 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=∑_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 X^j=(∑_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.]


page 1

page 2

page 3

page 4


Jackknife empirical likelihood based inference for Probability weighted moments

In the present article, we discuss jackknife empirical likelihood (JEL) ...

Networks of reinforced stochastic processes: estimation of the probability of asymptotic polarization

In a network of reinforced stochastic processes [arXiv:2206.07514, arXiv...

Consistency and asymptotic normality in a class of nearly unstable processes

This paper deals with inference in a class of stable but nearly-unstable...

Limit theorems for invariant distributions

We consider random processes whose distribution satisfies a symmetry pro...

Statistical Inference and A/B Testing for First-Price Pacing Equilibria

We initiate the study of statistical inference and A/B testing for first...

The Combinatorics of Barrier Synchronization

In this paper we study the notion of synchronization from the point of v...

Urn models with random multiple drawing and random addition

We consider an urn model with multiple drawing and random time-dependent...

Please sign up or login with your details

Forgot password? Click here to reset