eprintid: 3701
rev_number: 7
eprint_status: archive
userid: 36
dir: disk0/00/00/37/01
datestamp: 2017-05-08 12:57:15
lastmod: 2017-05-08 12:57:15
status_changed: 2017-05-08 12:57:15
type: monograph
metadata_visibility: show
creators_name: Aletti, Giacomo
creators_name: Crimaldi, Irene
creators_name: Ghiglietti, Andrea
creators_id: giacomo.aletti@unimi.it
creators_id: irene.crimaldi@imtlucca.it
creators_id: andrea.ghiglietti@unimi.it
title: Networks of reinforced stochastic processes: Asymptotics for the empirical means
ispublished: submitted
subjects: HA
subjects: QA
divisions: EIC
full_text_status: none
monograph_type: technical_report
keywords: Interacting Systems; Reinforced Stochastic Processes; Urn Models; Complex Networks; Synchronization; Asymptotic Normality.
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 not equal to 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. [2, 21]); while the asymptotic behavior of the stochastic
processes of the actions (X_{n,j})_n 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 converge and tools to
make inference on the matrix W.
date: 2017
date_type: submitted
id_number: arXiv:1705.02126
institution: IMT Institute for Advanced Studies Lucca
official_url: https://arxiv.org/abs/1705.02126
citation:   Aletti, Giacomo and Crimaldi, Irene and Ghiglietti, Andrea  Networks of reinforced stochastic processes: Asymptotics for the empirical means.  Technical Report       (Submitted)