Asymptotic behavior of interacting innovation processes

Ghiglietti Andrea, Università degli Studi di Milano-Bicocca

We introduce a new model based on ternary urns with triggering to incorporate different sources of interaction in a set of innovation processes. Any process is considered as located in a specfici vertex of a finite directed graph and each type of interaction is modeled by a dedicated weighted adjacency matrix. These urn models, known also as Poisson-Dirichlet processes, are characterized by a reinforcement mechanism such that the occurrence of a novelty or the selection of a specific item increases the probability of obseving the same event in the future. We study the asymptotic behavior of this set of innovation processes by proving first and second-order results on the most important quantities related with the dynamics of the system. In particular, we show that, for strongly connected graphs, the number of novelties arose in different vertices grow with a power-law having the same exponent and the ratio between them converges almost surely to a constant given by leading Perron eigenvector of one of the adjacency matrices. Moreover, the number of times each item is selected in the graph is uniformly distributed among the vertices. Regarding the second-order properties, we are able to establish the exact convergence rate and the corresponding limiting distribution of several quantities of interest. In particular, we prove joint central limit theorems of the entire system for the number of novelties observed and the number of times each item is selected in any vertex of the graph.

Area: CS40 - Recent developments for urn models (Andrea Ghiglietti and Giacomo Aletti)

Keywords: Dirichlet-Poisson innovation processes; Interacting random systems; Almost sure convergence and asymptotic normality

Please Login in order to download this file