A Strong Law for Some Generalized Urn Processes

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 $f$.