scholarly journals The dominating colour of an infinite Pólya urn model

2016 ◽  
Vol 53 (3) ◽  
pp. 914-924 ◽  
Author(s):  
Erik Thörnblad
Keyword(s):  
Random Graph ◽  
Finite Time ◽  
Classical Result ◽  
Graph Model ◽  
Urn Model ◽  
Random Graph Model ◽  
Pólya Urn ◽  
Polya Urn ◽  
Pólya Urn Model ◽  
Crucial Part

AbstractWe study a Pólya-type urn model defined as follows. Start at time 0 with a single ball of some colour. Then, at each time n≥1, choose a ball from the urn uniformly at random. With probability ½<p<1, return the ball to the urn along with another ball of the same colour. With probability 1−p, recolour the ball to a new colour and then return it to the urn. This is equivalent to the supercritical case of a random graph model studied by Backhausz and Móri (2015), (2016) and Thörnblad (2015). We prove that, with probability 1, there is a dominating colour, in the sense that, after some random but finite time, there is a colour that always has the most number of balls. A crucial part of the proof is the analysis of an urn model with two colours, in which the observed ball is returned to the urn along with another ball of the same colour with probability p, and removed with probability 1−p. Our results here generalise a classical result about the Pólya urn model (which corresponds to p=1).

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.

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 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.

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.

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