When are touchpoints limits for generalized Pólya urns?

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.