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Let f be a continuous function from the unit interval to itself and let X₀, X₁, be the successive proportions of red balls in an urn to which at the nth stage a red ball is added with probability f (Xₙ) and a black ball with probability 1 - f (Xₙ). Then Xₙ converges almost surely to a random variable X with support contained in the set C = \p: f (p) = p\. If, in addition, 0 0 (=0) when f' (r) 1). These results are extended to more general functions f.
Hill et al. (Tue,) studied this question.