Synchronization and fluctuation theorems for interacting Friedman urns
01 Dec 2016-Journal of Applied Probability (Applied Probability Trust)-Vol. 53, Iss: 4, pp 1221-1239
TL;DR: It is shown that the urns synchronize almost surely and that the fraction of balls of each colour converges to the deterministic limit of one-half, which matches with the limit known for a single Friedman urn.
Abstract: We consider a model of N interacting two-colour Friedman urns. The interaction model considered is such that the reinforcement of each urn depends on the fraction of balls of a particular colour in that urn as well as the overall fraction of balls of that colour in all the urns combined together. We show that the urns synchronize almost surely and that the fraction of balls of each colour converges to the deterministic limit of one-half, which matches with the limit known for a single Friedman urn. Furthermore, we use the notion of stable convergence to obtain limit theorems for fluctuations around the synchronization limit.
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01 Jan 2013
TL;DR: It is shown that the urns synchronizealmost surely, in the sense that the fraction of balls of a given color converges almost surely, as the time goes to infinity, to the same limit for all urneds.
Abstract: We consider a system of urns of Polya-type, with balls of two colors; the reinforcement of each urn depends both on the content of the same urn and on the average content of all urns. We show that the urns synchronize almost surely, in the sense that the fraction of balls of a given color converges almost surely, as the time goes to infinity, to the same limit for all urns. A normal approximation for a large number of urns is also obtained.
22 citations
TL;DR: In this paper, the Central Limit Theorem was obtained for a class of time-inhomogeneous interacting random walks on the simplex of probability measures over a finite set, where the increments of the walks are correlated, forcing their convergence to the same, possibly random, limit.
17 citations
TL;DR: In this article, a model of interacting stochastic processes with reinforcement has been introduced, where synchronization is induced along time by the reinforcement mechanism itself and does not require a large-scale limit.
Abstract: Randomly evolving systems composed by elements which interact among each other have always been of great interest in several scientific fields. This work deals with the synchronization phenomenon, that could be roughly defined as the tendency of different components to adopt a common behavior. We continue the study of a model of interacting stochastic processes with reinforcement, that recently has been introduced in Crimaldi et al. (2016, arXiv:1602.06217). Generally speaking, by reinforcement we mean any mechanism for which the probability that a given event occurs has an increasing dependence on the number of times that events of the same type occurred in the past. The particularity of systems of such stochastic processes is that synchronization is induced along time by the reinforcement mechanism itself and does not require a large-scale limit. We focus on the relationship between the topology of the network of the interactions and the long-time synchronization phenomenon. After proving the almost sure synchronization, we provide some CLTs in the sense of stable convergence that establish the convergence rates and the asymptotic distributions for both convergence to the common limit and synchronization. The obtained results lead to the construction of asymptotic confidence intervals for the limit random variable and of statistical tests to make inference on the topology of the network given the observation of the reinforced stochastic processes positioned at the vertices.
17 citations
Cites result from "Synchronization and fluctuation the..."
...In [22, 25, 43], the authors consider interacting urns (precisely, [22] and [25] deal with Pólya urns and [43] regards Friedman’s 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....
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TL;DR: In this paper, the authors studied the asymptotic behavior of the empirical means of the stochastic processes of the personal inclinations of the agents in the graph, proving their almost sure synchronization and central limit theorems in the sense of stable convergence.
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 $h
eq j$, 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. Aletti, Crimaldi, and Ghiglietti [arXiv:1607.08514, Ann. Appl. Probab., 27(6):3787-3844, 2017]; Crimaldi, Dai Pra, Louis, and Minelli [arXiv:1602.06217, Forthcoming in Stochastic Process. Appl.]); while the asymptotic behavior of quantities based on the stochastic processes $X^j$ of the actions 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 $W$.
11 citations
Additional excerpts
...[3, 10, 14, 16, 22, 25, 32, 33, 34, 37, 40, 43]) and their variants and generalizations (e....
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TL;DR: In this article, a new variant of the Eggenberger-polya urn, called the "Rescaled" polya, is introduced, which is characterized by the following features: (i) a local reinforcement mechanism mainly based on the last observations, (ii) a random persistent fluctuation of the predictive mean, and (iii) a long-term convergence of the empirical mean to a deterministic limit.
Abstract: Motivated by recent studies of big samples, this work aims at constructing a parametric model which is characterized by the following features: (i) a "local" reinforcement, i.e. a reinforcement mechanism mainly based on the last observations, (ii) a random persistent fluctuation of the predictive mean, and (iii) a long-term convergence of the empirical mean to a deterministic limit, together with a chi-squared goodness of fit result. This triple purpose has been achieved by the introduction of a new variant of the Eggenberger-Polya urn, that we call the "Rescaled" Polya urn. We provide a complete asymptotic characterization of this model, pointing out that, for a certain choice of the parameters, it has properties different from the ones typically exhibited from the other urn models in the literature. Therefore, beyond the possible statistical application, this work could be interesting for those who are concerned with stochastic processes with reinforcement.
9 citations
Cites background from "Synchronization and fluctuation the..."
...Finally, we observe that Equation (11) recalls the dynamics of a RSP with a “forcing input” (see [1, 19, 49]), but the main difference relies on the fact that, for the RP urn, the sequence (r∗ n) is such that r ∗ n → r∗ > 0, and so ∑ n 1/r ∗ n = +∞ and ∑ n 1/(r ∗ n) 2 = +∞, when β ∈ [0, 1), and such that ∑ n 1/r ∗ n < +∞ (and ∑ n 1/(r ∗ n) 2 < +∞) when β > 1....
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...2] and, as explained in details in Section 2, it does not belong to the class of Reinforced Stochastic Processes studied in [1, 3, 2, 19, 20, 22, 49]....
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References
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"Synchronization and fluctuation the..." refers result in this paper
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01 Sep 2008
TL;DR: In this article, the authors present a convergence analysis for lock-in probability, stability criteria, and synchronous schemes with different timescales, and a limit theorem for fluctuations.
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"Synchronization and fluctuation the..." refers background in this paper
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...Stochastic approximation approach Stochastic approximation schemes are heavily used in optimization problems and reinforcement learning (see [4])....
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TL;DR: The models surveyed in this paper include generalized Polya urns, reinforced random walks, interacting urn models, and continuous reinforced processes, with a focus on methods and results, with sketches provided of some proofs.
Abstract: The models surveyed include generalized Polya urns,
reinforced random walks, interacting urn models, and
continuous reinforced processes. Emphasis is on methods and
results, with sketches provided of some proofs. Applications
are discussed in statistics, biology, economics and a number
of other areas.
617 citations
"Synchronization and fluctuation the..." refers background or methods in this paper
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30 Jun 2008
TL;DR: In this paper, a collection of modern and evolving urn theory and its numerous applications are discussed, including exchangeability, stochastic processes via urns, and functional equations for moment generating functions can be obtained and solved.
Abstract: Incorporating a collection of recent results, Plya Urn Models deals with discrete probability through the modern and evolving urn theory and its numerous applications. The book first substantiates the realization of distributions with urn arguments and introduces several modern tools, including exchangeability and stochastic processes via urns. It reviews classical probability problems and presents dichromatic Plya urns as a basic discrete structure growing in discrete time. The author then embeds the discrete Plya urn scheme in Poisson processes to achieve an equivalent view in continuous time, provides heuristical arguments to connect the Plya process to the discrete urn scheme, and explores extensions and generalizations. He also discusses how functional equations for moment generating functions can be obtained and solved. The final chapters cover applications of urns to computer science and bioscience. Examining how urns can help conceptualize discrete probability principles, this book provides information pertinent to the modeling of dynamically evolving systems where particles come and go according to governing rules.
396 citations