Strong convergence of infinite color balanced urns under uniform ergodicity

Bandyopadhyay, Antar; Janson, Svante; Thacker, Debleena

doi:10.1017/jpr.2020.37

Cited by 9 publications

(12 citation statements)

References 36 publications

(78 reference statements)

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“…In Theorem 2.13(3), the situation is similar to Theorem 2.13 (2). The characterization is more technical, partly because the limit distribution now also depends on the initial values m 0 ; we give several equivalent conditions.…”

Section: Model and Main Resultsmentioning

confidence: 94%

“…and hence it is sufficient to prove (2.33) for f − Π ∆ ′ f instead of f . In other words, in (2) we may assume that…”

Section: Proof Of Theorem 210mentioning

confidence: 99%

“…In the analogue of the irreducible case, the theory is very recent and, as far as we know, there are only five papers on the subject: Bandyopadhyay and Thacker [3], Mailler and Marckert [26], Janson [21], Mailler and Villemonais [27], and Bandyopadhyay, Janson and Thacker [2]. The main difficulty is that the linear algebra tools used in the study of Pólya urns are replaced by operator theory in an infinite dimensional space.…”

mentioning

confidence: 99%

“…Bandyopadhyay, Janson and Thacker [2] later built on these methods to prove that the convergence results of [3] and [26] hold almost surely, under a condition that they call "uniform ergodicity" on the underlying Markov chain (w n ), and if the set of colours is countable.…”

mentioning

confidence: 99%

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Fluctuations of balanced urns with infinitely many colours

Janson¹,

Mailler²,

Villemonais³

2021

Preprint

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In this paper, we prove convergence and fluctuation results for measurevalued Pólya processes (MVPPs, also known as Pólya urns with infinitely-many colours). Our convergence results hold almost surely and in L 2 , under assumptions that are different from that of other convergence results in the literature. Our fluctuation results are the first second-order results in the literature on MVPPs; they generalise classical fluctuation results from the literature on finitely-many-colour Pólya urns. As in the finitely-many-colour case, the order and shape of the fluctuations depend on whether the "spectral gap is small or large".To prove these results, we show that MVPPs are stochastic approximations taking values in the set of measures on a measurable space E (the colour space). We then use martingale methods and standard operator theory to prove convergence and fluctuation results for these stochastic approximations.

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Section: Model and Main Resultsmentioning

confidence: 94%

“…and hence it is sufficient to prove (2.33) for f − Π ∆ ′ f instead of f . In other words, in (2) we may assume that…”

Section: Proof Of Theorem 210mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Fluctuations of balanced urns with infinitely many colours

Janson¹,

Mailler²,

Villemonais³

2021

Preprint

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show abstract

“…Urn models with infinitely many colors were treated in [5,6,18]. In [6] the authors introduce a class of balanced urn schemes with infinitely many colors indexed by Z d where the replacement schemes are given by the transition matrices associated with bounded increment random walks.…”

Section: Introductionmentioning

confidence: 99%

Positive reinforced generalized time-dependent Pólya urns via stochastic approximation

Ruszel¹,

Thacker²

2022

Preprint

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Consider a generalized time-dependent Pólya urn process defined as follows. Let d ∈ N be the number of urns/colors. At each time n, we distribute σn balls randomly to the d urns, proportionally to f , where f is a valid reinforcement function. We consider a general class of positive reinforcement functions R assuming some monotonicity and growth condition. The class R includes convex functions and the classical case f (x) = x α , α > 1. The novelty of the paper lies in extending stochastic approximation techniques to the d-dimensional case and proving that eventually the process will fixate at some random urn and the other urns will not receive any balls any more. x=1 1 f (x) < ∞.

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A new approach to Pólya urn schemes and its infinite color generalization

Bandyopadhyay

Thacker

2022

Ann. Appl. Probab.

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In this work, we introduce a generalization of the classical Pólya urn scheme [40] with colors indexed by a Polish space, say, S. The urns are defined as finite measures on S endowed with the Borel σ-algebra, say, S. The generalization is an extension of a model introduced earlier by Blackwell and MacQueen [8]. We present a novel approach of representing the observed sequence of colors from such a scheme in terms an associated branching Markov chain on the random recursive tree. The work presents fairly general asymptotic results for this new generalized urn models. As special cases we show that the results on classical urns, as well as, some of the results proved recently for infinite color urn models in [6,5], can easily be derived using the general asymptotic. We also demonstrate some newer results for infinite color urns.

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Strong convergence of infinite color balanced urns under uniform ergodicity

Cited by 9 publications

References 36 publications

Fluctuations of balanced urns with infinitely many colours

Fluctuations of balanced urns with infinitely many colours

Positive reinforced generalized time-dependent Pólya urns via stochastic approximation

A new approach to Pólya urn schemes and its infinite color generalization

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