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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


When are touchpoints limits for generalized Pólya urns?

Author: Robin Pemantle
Journal: Proc. Amer. Math. Soc. 113 (1991), 235-243
MSC: Primary 60F15; Secondary 60G42
MathSciNet review: 1055778
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Abstract: Hill, Lane, and Sudderth (1980) consider a Pólya-like urn scheme in which $ {X_0},{X_1}, \ldots $, are the successive proportions of red balls in an urn to which at the $ n$ th stage a red ball is added with probability $ f({X_n})$ and a black ball is added with probability $ 1 - f({X_n})$. For continuous $ f$ they show that $ {X_n}$ converges almost surely to a random limit $ X$ which is a fixed point for $ f$ and ask whether the point $ p$ can be a limit if $ p$ is a touchpoint, i.e. $ p = f(p)$ but $ f(x) > x$ for $ x \ne p$ in a neighborhood of $ p$. The answer is that it depends on whether the limit of $ (f(x) - x)/(p - x)$ is greater or less than 1/2 as $ x$ approaches $ p$ from the side where $ (f(x) - x)/(p - x)$ is positive.

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PII: S 0002-9939(1991)1055778-8
Keywords: Pólya urn, touchpoint
Article copyright: © Copyright 1991 American Mathematical Society