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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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When are touchpoints limits for generalized Pólya urns?
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by Robin Pemantle
Proc. Amer. Math. Soc. 113 (1991), 235-243
DOI: https://doi.org/10.1090/S0002-9939-1991-1055778-8

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
  • 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
  • R. Pemantle, Random processes with reinforcement, Doctoral thesis, Massachusetts Institute of Technology, 1988.
  • 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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Bibliographic Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 113 (1991), 235-243
  • MSC: Primary 60F15; Secondary 60G42
  • DOI: https://doi.org/10.1090/S0002-9939-1991-1055778-8
  • MathSciNet review: 1055778