A central limit theorem, and related results, for a two-color randomly reinforced urn

Aletti, Giacomo; May, Caterina; Secchi, Piercesare

doi:10.1239/aap/1253281065

Cited by 24 publications

(48 citation statements)

References 18 publications

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“…Hence, according to [23], we say that Y n converges to K stably in the strong sense, with respect to G = (G n ) n , if Finally, a strengthening of the stable convergence in the strong sense can be naturally obtained if in (80) we replace the convergence in probability by the almost sure convergence: given a conditioning system G = (G n ) n , we say that Y n converges to K in the sense of the almost sure conditional convergence, with respect to G, if E [f (Y n ) | G n ] a.s. −→ Kf for each bounded continuous real function f on S. The almost sure conditional convergence has been introduced in [18] and, subsequently, employed by others in the urn model literature (e.g. [6,45]).…”

Section: Appendix Appendix a Some Technical Resultsmentioning

confidence: 99%

Networks of reinforced stochastic processes: Asymptotics for the empirical means

Aletti¹,

Crimaldi²,

Ghiglietti³

2019

Bernoulli

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This work deals with systems of interacting reinforced stochastic processes, where each process X j = (Xn,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" Zn,j and on the inclinations Z n,h , with h = j, of the other agents according to the elements of W .Asymptotic results for the stochastic processes of the personal inclinations Z j = (Zn,j )n have been subject of studies in recent papers (e.g. [2,21]); 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 Nn,j = n k=1 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 .

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Section: Appendix Appendix a Some Technical Resultsmentioning

confidence: 99%

Networks of reinforced stochastic processes: Asymptotics for the empirical means

Aletti¹,

Crimaldi²,

Ghiglietti³

2019

Bernoulli

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“…Remark A. 3. We note that, if γ = 1 and (68) holds, then we can add to (73) the following: if uc(a 1 + a 2 ) < (2u − 1).…”

Section: −→ Qmentioning

confidence: 99%

Synchronization of reinforced stochastic processes with a network-based interaction

Aletti¹,

Crimaldi²,

Ghiglietti³

2017

Ann. Appl. Probab.

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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 [12]. 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.

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“…Pitman and Yor (1997) and Pitman (2006, Section 3.2). The case α = 0 essentially reduces to the randomly reinforced urn model that has been deeply studied by many authors (see, for instance, Aletti et al (2009) May et al (2005), Pemantle (2007), and the references therein; see also Section 5 of this paper). The case when µ is discrete and α > 0 has been treated in Berti et al (2009).…”

Section: Generalized Poisson-dirichlet Sequencesmentioning

confidence: 99%

“…Taking into account the analogy with randomly reinforced Pólya urns, it is natural to think that the random variable V A is generally not degenerate (see Aletti et al (2009)). This fact implies that A is not degenerate, while U A is 0 if and only if Y n converges in L 2 to the constant m. This happens, for example, in the classical case (see (4.1)) studied in Pitman and Yor (1997) and Pitman (2006, Section 3.2).…”

Section: Case M > αmentioning

confidence: 99%

“…For instance, instead of a classical Pólya urn scheme, it may be useful to deal with the so-called randomly reinforced urn scheme. See, for example, Aletti et al (2009), Bai and Hu (2005), Berti et al (2004), Berti et al (2009), Crimaldi (2009, Crimaldi and Leisen (2008), Flournoy (2009), Janson (2006), May et al (2005), Pemantle (2007), and the references therein. Such processes fail to be exchangeable.…”

Section: Introductionmentioning

confidence: 99%

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Conditionally identically distributed species sampling sequences

2010

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In this paper the theory of species sampling sequences is linked to the theory of conditionally identically distributed sequences in order to enlarge the set of species sampling sequences which are mathematically tractable. The conditional identity in distribution (see Berti, Pratelli and Rigo (2004)) is a new type of dependence for random variables, which generalizes the well-known notion of exchangeability. In this paper a class of random sequences, called generalized species sampling sequences, is defined and a condition to have conditional identity in distribution is given. Moreover, two types of generalized species sampling sequence that are conditionally identically distributed are introduced and studied: the generalized Poisson-Dirichlet sequence and the generalized Ottawa sequence. Some examples are discussed.

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A central limit theorem, and related results, for a two-color randomly reinforced urn

Abstract: We prove a Central Limit Theorem for the sequence of random compositions of a two-color randomly reinforced urn. As a consequence, we are able to show that the distribution of the urn limit composition has no point masses.

Cited by 24 publications

References 18 publications

Networks of reinforced stochastic processes: Asymptotics for the empirical means

Networks of reinforced stochastic processes: Asymptotics for the empirical means

Synchronization of reinforced stochastic processes with a network-based interaction

Conditionally identically distributed species sampling sequences

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