Advances in Urn Models during the Past Two Decades

Kotz, Samuel; Balakrishnan, N.

doi:10.1007/978-1-4612-4140-9_14

Cited by 71 publications

(64 citation statements)

References 95 publications

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“…Using (16) and (17) we finally obtain First we consider the case m, n → ∞ such that m a/d = o(n), and directly obtain…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Urn models are simple, useful mathematical tools for describing many evolutionary processes in diverse fields of application such as analysis of algorithms and data structures, statistics and genetics. Due to their importance in applications, there is a huge literature on the stochastic behavior of urn models; see for example [11,16]. Recently, a few different approaches have been proposed, which yield deep and far-reaching results for very general urn models; see [3,4,9,10].…”

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confidence: 99%

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Limiting Distributions for a Class Of Diminishing Urn Models

Kuba

Panholzer

2012

Adv. Appl. Probab.

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ABSTRACT. In this work we analyze a class of 2 × 2 Pólya-Eggenberger urn models with ball replacement matrix M = −a 0 c −d , a, d ∈ N, and c = p · a with p ∈ N 0 . We obtain limiting distributions for this 2 × 2 urn model by obtaining a precise recursive description of the moments of the considered random variables, which allows us to deduce asymptotic expansions of the moments. In particular, we obtain limiting distributions for the pills problem a = c = d = 1, originally proposed by Knuth and McCarthy. Furthermore, we also obtain limiting distributions for the well known sampling without replacement urn, a = d = 1 and c = 0, and generalizations of it to arbitrary a, d ∈ N and c = 0. Moreover, we obtain a recursive description of the moment sequence for a generalized problem.

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“…Using (16) and (17) we finally obtain First we consider the case m, n → ∞ such that m a/d = o(n), and directly obtain…”

Section: Proof Of Theoremmentioning

confidence: 99%

mentioning

confidence: 99%

Limiting Distributions for a Class Of Diminishing Urn Models

Kuba

Panholzer

2012

Adv. Appl. Probab.

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“…In these classical models, square ball replacement matrices underly the random structures, and their eigenvalues play a significant role in the formulation of asymptotic results. For background see [9], and [11]- [13]. In recent years, several new theoretical studies and applications required the consideration of models with multiple drawing (drawing multiple balls each time).…”

Section: Introductionmentioning

confidence: 99%

Analysis of a generalized Friedman’s urn with multiple drawings

Kuba

Mahmoud

Panholzer

2013

Discrete Applied Mathematics

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ABSTRACT. We study a generalized Friedman's urn model with multiple drawings of white and blue balls. After a drawing, the replacement follows a policy of opposite reinforcement. We give the exact expected value and variance of the number of white balls after a number of draws, and determine the structure of the moments. Moreover, we obtain a strong law of large numbers, and a central limit theorem for the number of white balls. Interestingly, the central limit theorem is obtained combinatorially via the method of moments and probabilistically via martingales. We briefly discuss the merits of each approach. The connection to a few other related urn models is briefly sketched.

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“…The advent of Dirichlet limits, when G is chosen appropriately, seems of particular interest, given similar results for limit color-frequencies in Pólya urns [4], [10], as it hints at an even larger role for Dirichlet measures in related but different "reinforcement"-type models (see [17], [23], [22], and references therein, for more on urn and reinforcement schemes). In this context, the set of "spreading" limits µ G in Theorem 1.3, in which Dirichlet measures are but a subset, appears intriguing as well (cf.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Occupation laws for some time-nonhomogeneous Markov chains

Dietz

Sethuraman

2007

Electron. J. Probab.

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We consider finite-state time-nonhomogeneous Markov chains whose transition matrix at time n is I + G/n ζ where G is a "generator" matrix, that is G(i, j) > 0 for i, j distinct, and G(i, i) = − k =i G(i, k), and ζ > 0 is a strength parameter. In these chains, as time grows, the positions are less and less likely to change, and so form simple models of age-dependent time-reinforcing schemes. These chains, however, exhibit some different, perhaps unexpected, occupation behaviors depending on parameters.Although it is shown, on the one hand, that the position at time n converges to a point-mixture for all ζ > 0, on the other hand, the average occupation vector up to time n, when variously 0 < ζ < 1, ζ > 1 or ζ = 1, is seen to converge to a constant, a pointmixture, or a distribution µ G with no atoms and full support on a simplex respectively, as n ↑ ∞. This last type of limit can be interpreted as a sort of "spreading" between the cases 0 < ζ < 1 and ζ > 1.In particular, when G is appropriately chosen, intriguingly, µ G is a Dirichlet distribution, reminiscent of results in Pólya urns.

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Advances in Urn Models during the Past Two Decades

Cited by 71 publications

References 95 publications

Limiting Distributions for a Class Of Diminishing Urn Models

Limiting Distributions for a Class Of Diminishing Urn Models

Analysis of a generalized Friedman’s urn with multiple drawings

Occupation laws for some time-nonhomogeneous Markov chains

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