Open Access
April, 1980 A Strong Law for Some Generalized Urn Processes
Bruce M. Hill, David Lane, William Sudderth
Ann. Probab. 8(2): 214-226 (April, 1980). DOI: 10.1214/aop/1176994772


Let $f$ be a continuous function from the unit interval to itself and let $X_0, X_1, \cdots$ be 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 with probability $1 - f(X_n)$. Then $X_n$ converges almost surely to a random variable $X$ with support contained in the set $C = \{p: f(p) = p\}$. If, in addition, $0 < f(p) < 1$ for all $p$, then, for each $r$ in $C, P\lbrack X = r\rbrack > 0(=0)$ when $f'(r) < 1(> 1)$. These results are extended to more general functions $f$.


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Bruce M. Hill. David Lane. William Sudderth. "A Strong Law for Some Generalized Urn Processes." Ann. Probab. 8 (2) 214 - 226, April, 1980.


Published: April, 1980
First available in Project Euclid: 19 April 2007

zbMATH: 0429.60021
MathSciNet: MR566589
Digital Object Identifier: 10.1214/aop/1176994772

Primary: 60F15
Secondary: 60G17 , 60J05

Keywords: Polya process , split process , strategy , urn function , urn process

Rights: Copyright © 1980 Institute of Mathematical Statistics

Vol.8 • No. 2 • April, 1980
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