A Strong Law for Some Generalized Urn Processes

Bruce M. Hill; David A. Lane; William D. Sudderth

doi:10.1214/aop/1176994772

Abstract

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 $f$.

Keywords

MathematicsCombinatoricsRandom variableProbability theoryBall (mathematics)Law of large numbersUnit sphereMathematical analysisStatistics

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Publication Info

Year: 1980
Type: article
Volume: 8
Issue: 2
Citations: 174
Access: Closed

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APA Style

                            
                                    Bruce M. Hill, 
                                
                                    David A. Lane, 
                                
                                    William D. Sudderth
                                
                            (1980). 
                            A Strong Law for Some Generalized Urn Processes. 
                            The Annals of Probability
                            , 8
                            (2)
                            .
                            https://doi.org/10.1214/aop/1176994772

Identifiers

DOI: 10.1214/aop/1176994772