Linear de-preferential urn models

Bandyopadhyay, Antar; Kaur, Gursharn

doi:10.1017/apr.2018.55

Cited by 5 publications

(2 citation statements)

References 52 publications

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“…In this subsection, we review the strong law of large numbers and the central limit theorem on a version of uniform urn models developed by Choi et al (2021), which will be applied to the nonuniform urn process in Section 2.2 using the urn coupling idea in Bandyopadhyay and Kaur (2018).…”

Section: Limiting Theorems On Uniform Urn Modelsmentioning

confidence: 99%

Distributions of cherries and pitchforks for the Ford model

Kaur

Choi

2023

Theoretical Population Biology

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Section: Limiting Theorems On Uniform Urn Modelsmentioning

confidence: 99%

Distributions of cherries and pitchforks for the Ford model

Kaur

Choi

2023

Theoretical Population Biology

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“…In this subsection, we recall the strong laws of large numbers and the central limit theorems on a version of uniform urn models developed in [7], which will be related to the nonuniform urn process in Subsection 2.2 later using the urn coupling idea in [4].…”

Section: Limiting Theorems On Uniform Urn Modelsmentioning

confidence: 99%

Distributions of cherries and pitchforks for the Ford model

Kaur¹,

Choi²,

Wu³

2021

Preprint

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Distributional properties of tree shape statistics under random phylogenetic tree models play an important role in investigating evolutionary forces underlying real world phylogenies. In this paper, we study two subtree counting statistics, the number of cherries and that of pitchforks for Ford's alpha model, a one-parameter family of random phylogenetic tree models which includes as specific instances of both the uniform and the Yule models, two tree models commonly used in phylogenetics. Based on a version of the extended Pólya urn models, in which negative entries are permitted for their replacement matrices, we obtain the strong laws of large numbers and the central limit theorems for the joint distribution of these two count statistics for the Ford model. Furthermore, we derive a recursive formula for computing the exact joint distribution of these two statistics, which leads to higher order asymptotic expansions of their marginal and joint moments.

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Negatively reinforced balanced urn schemes

Kaur

2019

Advances in Applied Mathematics

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We consider weighted negatively reinforced urn schemes with finitely many colours. An urn scheme is called negatively reinforced, if the selection probability for a colour is proportional to the weight w of the colour proportion, where w is a non-increasing function. Under certain assumptions on the replacement matrix R and weight function w, such as, w is differentiable and w(0) < ∞, we obtain almost sure convergence of the random configuration of the urn model. In particular, we show that if R is doubly stochastic the random configuration of the urn converges to the uniform vector, and asymptotic normality holds, if the number of colours in the urn are sufficiently large.

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Linear de-preferential urn models

Cited by 5 publications

References 52 publications

Distributions of cherries and pitchforks for the Ford model

Distributions of cherries and pitchforks for the Ford model

Distributions of cherries and pitchforks for the Ford model

Negatively reinforced balanced urn schemes

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