Fluctuation theorems for synchronization of interacting Pólya’s urns

Abstract

We consider a model of N two-colors urns in which the reinforcement of each urn depends also on the content of all the other urns. This interaction is of mean-field type and it is tuned by a parameter α ∈ [0, 1]; in particular, for α = 0 the N urns behave as N independent Pólya's urns. For α > 0 urns synchronize, in the sense that the fraction of balls of a given color converges a.s. to the same (random) limit in all urns. In this paper we study fluctuations around this synchronized regime. The scaling of these fluctuations depends on the parameter α. In particular the standard scaling t −1/2 appears only for α > 1/2. For α ≥ 1/2 we also determine the limit distribution of the rescaled fluctuations. We use the notion of stable convergence, which is stronger than convergence in distribution.

Synchronization of reinforced stochastic processes with a network-based interaction

“…The present work have some issues in common with [12,13] and [2], but at the same time some significant differences can be pointed out. In particular, we share with [2] a general interacting framework driven by the interacting matrix (here called weighted adjacency matrix).…”

“…The main results can be informally stated as follows: if p ≥ 1/2, then all the urns fixate on the same color after a finite time, and if p < 1/2, then some urns fixate on a unique color and others keep drawing both colors. In [13,16,32] the authors consider interacting urns (precisely, [13] and [16] deal with Pólya urns and [32] regards Friedman urns) in which the interaction can be defined again as of the mean-field type, but the reinforcement scheme is different from the previous one: indeed, the urns interact among each other through the average composition in the entire system, tuned by the interaction parameter α, and the probability of drawing a ball of a certain color is proportional to the number of balls of that color, rather than to its exponential, leading to quite different results. Synchronization and central limit theorems for the urn proportions have been proven for different values of the tuning parameter α, providing different convergence rates and asymptotic variances.…”

“…This class does not include the generalized Friedman urns studied in [2]; while it includes urn models with not-irreducible mean replacement matrices, as Pólya urns. With [12] we share the class of reinforced stochastic processes considered, which contain Pólya urns also studied in [13]. However, with respect to [12,13], we generalize the form of interaction since here we deal with a general weighted adjacency matrix instead of just the mean-field interaction.…”

Networks of reinforced stochastic processes: Asymptotics for the empirical means

“…An interesting example of interacting system is provided by the "mean-field interaction", already considered in [2,21,22,25]. Naturally, all the weighted adjacency matrices introduced and analyzed in [2] can be considered as well.…”

“…The previous quoted papers [2,21,22,25] are all focused on the asymptotic behavior of the stochastic processes of the "personal inclinations" {Z j = (Z n,j ) n : j ∈ V } of the agents. On the contrary, in this work we focus on the average of times in which the agents adopt "action 1", i.e.…”