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Aletti, Giacomo; Crimaldi, Irene; Ghiglietti, Andrea

Networks of reinforced stochastic processes: asymptotics for the empirical means. (English) Zbl 1430.60078

Bernoulli 25, No. 4B, 3339-3378 (2019).
Consider a network of \(N\) interacting stochastic processes, positioned at the vertices of an underlying graph governing which processes interact directly. The processes at each vertex take values in \(\{0,1\}\), updating at each time \(n\) according to conditionally independent Bernoulli random variables with parameters which are linear combinations of “inclination” random variables \(Z_{n,j}\) at time \(n\) for those vertices \(j\) with which a given vertex interacts directly. The authors prove limiting results for the empirical means of these stochastic processes, that is, asymptotics for the proportion of time each process spends in the state \(\{1\}\). Their first result shows that these empirical means all converge almost surely to the same random variable, \(Z_\infty\). To complement this result, the authors also prove joint central limit theorems for these empirical means and the “inclination” variables. The covariance structures of the limiting Gaussian random variables obtained in these results depends on the asymptotic behaviour of a sequence governing the update of the “inclination” random variables at each time step, the underlying graph, and the random variable \(Z_\infty\). The authors also discuss some statistical applications of their results, such as the construction of asymptotic confidence intervals for statistics of interest and inference on the underlying graph structure.
Reviewer: Fraser Daly (Edinburgh)

Cited in 1 Review
Cited in 8 Documents

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F05 Central limit and other weak theorems
60F15 Strong limit theorems
62E20 Asymptotic distribution theory in statistics

Keywords:

asymptotic normality; complex networks; interacting systems; reinforced stochastic processes; synchronization; urn models; central limit theorem; almost sure convergence
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\textit{G. Aletti} et al., Bernoulli 25, No. 4B, 3339--3378 (2019; Zbl 1430.60078)
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