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 , … {X_0},{X_1}, \ldots , are the successive proportions of red balls in an urn to which at the n n th stage a red ball is added with probability f ( X n ) f({X_n}) and a black ball is added with probability 1 − f ( X n ) 1 - f({X_n}) . For continuous f f they show that X n {X_n} converges almost surely to a random limit X X which is a fixed point for f f and ask whether the point p p can be a limit if p p is a touchpoint, i.e. p = f ( p ) p = f(p) but f ( x ) > x f(x) > x for x ≠ p x \ne p in a neighborhood of p p . The answer is that it depends on whether the limit of ( f ( x ) − x ) / ( p − x ) (f(x) - x)/(p - x) is greater or less than 1/2 as x x approaches p p from the side where ( f ( x ) − x ) / ( p − x ) (f(x) - x)/(p - x) is positive.