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
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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
- 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
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