A time-dependent version of Pólya's urn
type: Item
instance of: scholarly article
title: A time-dependent version of Pólya's urn
zbMATH Open document ID: 0708.60015
DOI: 10.1007/BF01046101
author: Robin Pemantle
published in: Journal of Theoretical Probability
publication date: 1990-01-01T00:00:00Z
review text: A Pólya urn is considered with the modification that the number of extra balls added of the color drawn is a function of time. An urn containing R red balls and B black balls at time \(n=1\) is considered. A ball is drawn at random from the urn and replaced along with another ball of the same color, so that at time \(n=2\) there are \(R+B+1\) balls; \(R+1\) of them are red with \(R/(R+B)\) and R of them are red otherwise. Draw and replacement are repeated ad infinitum, with the probability of drawing a red ball always equal to the proportion of balls in the urn that are red at that time. The proportions of red balls converge almost surely to a limit that is random and has beta distribution with parameters R and B. The author shows that the proportion of red balls in the urn converges to a random limit that may have a nonzero probability or being 0 or 1, but is nonatomic elsewhere.
Mathematics Subject Classification ID: 60C05
Mathematics Subject Classification ID: 60E05
zbMATH DE Number: 4163806
zbMATH Keywords: Pólya urn
reviewed by: Gabriel V. Orman
MaRDI profile type: Publication
cites work: A generalized URN problem and its applications
cites work: Q5578191
cites work: Q5512461
cites work: A simple urn model
cites work: A strong law for some generalized urn processes
cites work: Phase transition in reinforced random walk and RWRE on trees
cites work: Nonconvergence to unstable points in urn models and stochastic approximations
full work available at URL: https://doi.org/10.1007/bf01046101
OpenAlex ID: W1997333829
description: scientific article; zbMATH DE number 4163806