Strong convergence of proportions in a multicolor Pólya urn

Gouet, Raúl

doi:10.2307/3215382

Cited by 40 publications

(62 citation statements)

References 8 publications

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“…It is worthwhile to note here that for S finite our result is essentially the classical result for Freedman-Pólya-Eggenberger-type urn models [2,3,18,24]. The classical results mainly use three types of technique, namely martingale techniques [8,9,12,18], stochastic approximations [28], and embedding into continuous-time pure birth processes [2,3,24,25]. Typically the analysis of a finite color urn is heavily dependent on the Perron-Frobenius theory [36] of matrices with positive entries and the Jordan decomposition of finite-dimensional matrices [2,3,8,12,18,24,25].…”

Section: Background and Motivationsupporting

confidence: 70%

“…As discussed above, the two assumptions are identical when S is finite. It is worthwhile to note here that for S finite our result is essentially the classical result for Freedman-Pólya-Eggenberger-type urn models [2,3,18,24]. The classical results mainly use three types of technique, namely martingale techniques [8,9,12,18], stochastic approximations [28], and embedding into continuous-time pure birth processes [2,3,24,25].…”

Section: Background and Motivationmentioning

confidence: 68%

“…The classical results mainly use three types of technique, namely martingale techniques [8,9,12,18], stochastic approximations [28], and embedding into continuous-time pure birth processes [2,3,24,25]. Typically the analysis of a finite color urn is heavily dependent on the Perron-Frobenius theory [36] of matrices with positive entries and the Jordan decomposition of finite-dimensional matrices [2,3,8,12,18,24,25]. Unfortunately such techniques are unavailable when S is infinite, even when countable.…”

Section: Background and Motivationmentioning

confidence: 99%

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Strong convergence of infinite color balanced urns under uniform ergodicity

2020

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We consider the generalization of the Pólya urn scheme with possibly infinitely many colors, as introduced in [37], [4], [5], and [6]. For countably many colors, we prove almost sure convergence of the urn configuration under the uniform ergodicity assumption on the associated Markov chain. The proof uses a stochastic coupling of the sequence of chosen colors with a branching Markov chain on a weighted random recursive tree as described in [6], [31], and [26]. Using this coupling we estimate the covariance between any two selected colors. In particular, we re-prove the limit theorem for the classical urn models with finitely many colors.

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Section: Background and Motivationsupporting

confidence: 70%

Section: Background and Motivationmentioning

confidence: 68%

Section: Background and Motivationmentioning

confidence: 99%

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Strong convergence of infinite color balanced urns under uniform ergodicity

2020

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“…Urn schemes, which were first studied by Pólya [48], are perhaps the simplest reinforcement models. They have many applications and generalizations [6]- [8], [11]- [13], [16], [17], [20], [21], [24], [25], [30]- [33], [35], [36], [39], [40], [45], [46], [48]. In general, reinforcement models typically adhere to the structure of 'rich get richer', which has also been termed positive reinforcement.…”

Section: Background and Motivationmentioning

confidence: 99%

Linear de-preferential urn models

Bandyopadhyay

Kaur

2018

Adv. Appl. Probab.

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In this paper we consider a new type of urn scheme, where the selection probabilities are proportional to a weight function, which is linear but decreasing in the proportion of existing colours. We refer to it as the de-preferential urn scheme. We establish the almost-sure limit of the random configuration for any balanced replacement matrix R. In particular, we show that the limiting configuration is uniform on the set of colours if and only if R is a doubly stochastic matrix. We further establish the almost-sure limit of the vector of colour counts and prove central limit theorems for the random configuration as well as for the colour counts.

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“…The first eigenvalue λ of the design matrix and its normalized first left (row) eigenvector v play a key role in the asymptotic properties of the GPU design. Many authors (for instance Athreya and Karlin, 1967; Gouet, 1997; Janson, 2004) studied asymptotic properties of X n and N ( n ) and proved that…”

Section: Description Of the Optimal Adaptive Gpu Designmentioning

confidence: 99%

Compound adaptive GPU design for clinical trials

Yuan

Bezandry

Bonney

2010

Journal of Statistical Planning and Inference

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In clinical trials, several competing treatments are often carried out in the same trial period. The goal is to assess the performances of these different treatments according to some optimality criterion and minimize risks to the patients in the entire process of the study. For this, each coming patient is allocated sequentially to one of the treatments according to a mechanism defined by the optimality criterion. In practice, sometimes different optimality criteria, or the same criterion with different regimes, need to be considered to assess the treatments in the same study, so that each mechanism is also evaluated through the trail study. In this case, the question is how to allocate the treatments to the incoming patients so that the criteria/mechanisms of interest are assessed during the trail process, and the overall performance of the trial is optimized under the combined criteria or regimes. In this paper, we consider this problem by investigating a compound adaptive generalized Pólya urn design. Basic asymptotic properties of this design are also studied.

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Strong convergence of proportions in a multicolor Pólya urn

Abstract: We prove strong convergence of the proportions Un/Tn of balls in a multitype generalized Pólya urn model, using martingale arguments. The limit is characterized as a convex combination of left dominant eigenvectors of the replacement matrix R, with random Dirichlet coefficients.

Cited by 40 publications

References 8 publications

Strong convergence of infinite color balanced urns under uniform ergodicity

Strong convergence of infinite color balanced urns under uniform ergodicity

Linear de-preferential urn models

Compound adaptive GPU design for clinical trials

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