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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.

References [Enhancements On Off] (What's this?)

  • [HLS] Bruce M. Hill, David Lane, and William Sudderth, A strong law for some generalized urn processes, Ann. Probab. 8 (1980), no. 2, 214–226. MR 566589
  • [P1] R. Pemantle, Random processes with reinforcement, Doctoral thesis, Massachusetts Institute of Technology, 1988.
  • [P2] Robin Pemantle, Nonconvergence to unstable points in urn models and stochastic approximations, Ann. Probab. 18 (1990), no. 2, 698–712. MR 1055428

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Keywords: Pólya urn, touchpoint
Article copyright: © Copyright 1991 American Mathematical Society