A generalized Pólya urn model and related multivariate distributions

2005 ◽  
Vol 57 (1) ◽  
pp. 49-59 ◽  
Author(s):  
Kiyoshi Inoue ◽  
Sigeo Aki
Keyword(s):  
Urn Model ◽  
Multivariate Distributions ◽  
Pólya Urn ◽  
Polya Urn ◽  
Pólya Urn Model ◽  
Generalized Pólya Urn

Strong convergence of proportions in a multicolor Pólya urn

1997 ◽  
Vol 34 (2) ◽  
pp. 426-435 ◽  
Author(s):  
Raúl Gouet
Keyword(s):  
Strong Convergence ◽  
Convex Combination ◽  
Urn Model ◽  
Pólya Urn ◽  
Polya Urn ◽  
Pólya Urn Model ◽  
Generalized Pólya Urn

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.

Strong convergence of proportions in a multicolor Pólya urn

1997 ◽  
Vol 34 (02) ◽  
pp. 426-435 ◽  
Author(s):  
Raúl Gouet
Keyword(s):  
Strong Convergence ◽  
Convex Combination ◽  
Urn Model ◽  
Pólya Urn ◽  
Polya Urn ◽  
Pólya Urn Model ◽  
Generalized Pólya Urn

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.

scholarly journals On Generalized Pólya Urn Models

2013 ◽  
Vol 50 (4) ◽  
pp. 1169-1186 ◽  
Author(s):  
May-Ru Chen ◽  
Markus Kuba
Keyword(s):  
Discrete Time ◽  
Higher Moments ◽  
Urn Models ◽  
Urn Model ◽  
Time Step ◽  
Model Case ◽  
Pólya Urn ◽  
Polya Urn ◽  
Generalized Pólya Urn

We study an urn model introduced in the paper of Chen and Wei (2005), where at each discrete time step m balls are drawn at random from the urn containing colors white and black. Balls are added to the urn according to the inspected colors, generalizing the well known Pólya-Eggenberger urn model, case m = 1. We provide exact expressions for the expectation and the variance of the number of white balls after n draws, and determine the structure of higher moments. Moreover, we discuss extensions to more than two colors. Furthermore, we introduce and discuss a new urn model where the sampling of the m balls is carried out in a step-by-step fashion, and also introduce a generalized Friedman's urn model.

Joint distributions of circular runs of various lengths based on multi-colour Pólya urn model

2012 ◽  
Vol 49 (3) ◽  
pp. 283-300
Author(s):  
Sonali Bhattacharya
Keyword(s):  
Probability Generating Function ◽  
Discrete Distribution ◽  
Time Sharing ◽  
Urn Model ◽  
Joint Distributions ◽  
Pólya Urn ◽  
Start Up ◽  
Polya Urn ◽  
Pólya Urn Model ◽  
First Time

In this paper, we have used Eryilmaz’s (2008) multi-colour Pólya urn model to obtain joint distributions of runs of t-types of exact lengths (k1, k2, …, kt), at least lengths (k1, k2, …, kt), non-overlapping runs of lengths (k1, k2, … kt) and overlapping runs of lengths (k1, k2, … kt) when counting of runs is done in a circular setup. We have also derived joint distributions of longest runs of various types under similar conditions. Distributions of runs have found applications in fields of reliability of consecutive-k-out-of n: F system, consecutive k-out-of-r-from n: F system, start-up demonstration test, molecular biology, radar detection, time sharing systems and quality control. The literature is profound in discussion of marginal distribution and joint distribution of runs of various types under linear and circular setup using techniques like urn model with balls of two or more colours, probability generating function and compounding discrete distribution with suitable beta functions. Through this paper for first time effort been made to discuss joint distributions of runs of various lengths and types using Multi-colour urn model.

A TIME-DEPENDENT PÓLYA URN WITH MULTIPLE DRAWINGS

2019 ◽  
Vol 34 (4) ◽  
pp. 469-483
Author(s):  
May-Ru Chen
Keyword(s):  
Random Variable ◽  
Time Dependent ◽  
Necessary And Sufficient Condition ◽  
Urn Model ◽  
Absolutely Continuous ◽  
Pólya Urn ◽  
Polya Urn ◽  
Generalized Pólya Urn ◽  
Necessary And Sufficient ◽  
Positive Integers

In this paper, we consider a generalized Pólya urn model with multiple drawings and time-dependent reinforcements. Suppose an urn initially contains w white and r red balls. At the nth action, m balls are drawn at random from the urn, say k white and m−k red balls, and then replaced in the urn along with cnk white and cn(m − k) red balls, where {cn} is a given sequence of positive integers. Repeat the above procedure ad infinitum. Let Xn be the proportion of the white balls in the urn after the nth action. We first show that Xn converges almost surely to a random variable X. Next, we give a necessary and sufficient condition for X to have a Bernoulli distribution with parameter w/(w + r). Finally, we prove that X is absolutely continuous if {cn} is bounded.

Bandwidth Expansionwith a pólya URN Model

2007 ◽  
Author(s):  
Bhiksha Raj ◽  
Rita Singh ◽  
Madhusudana Shashanka ◽  
Paris Smaragdis
Keyword(s):  
Urn Model ◽  
Pólya Urn ◽  
Polya Urn ◽  
Pólya Urn Model

A Pólya urn model and the coalescent

1992 ◽  
Vol 29 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Gudrun Trieb
Keyword(s):  
Population Genetics ◽  
Urn Model ◽  
Ewens Sampling Formula ◽  
Sampling Formula ◽  
Pólya Urn ◽  
Polya Urn ◽  
Pólya Urn Model

In recent papers by Hoppe and Donnelly it has been shown that a Pólya urn model generating the Ewens sampling formula (population genetics) parallels a construction of Kingman using a Poisson–Dirichlet ‘paintbox'. Even the jump chain of Kingman's n-coalescent can be constructed using the urn. The properties of a certain process based on the coalescent also are derived. This process was introduced by Hoppe.

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