A functional equation whose unknown is P([0,1]) valued

Details A functional equation whose unknown is P([0,1]) valued A functional equation whose unknown is P([0,1]) valued

2009-05-20

arxiv.org/abs/0905.3310v3

Mathematics

Probability

31 pages, pre-galleys version of accepted paper

Scientific paper

10.1007/s10959-011-0399-7

We study a functional equation whose unknown maps a Euclidean space into the space of probability distributions on [0,1]. We prove existence and uniqueness of its solution under suitable regularity and boundary conditions, we show that it depends continuously on the boundary datum, and we characterize solutions that are diffuse on [0,1]. A canonical solution is obtained by means of a Randomly Reinforced Urn with different reinforcement distributions having equal means. The general solution to the functional equation defines a new parametric collection of distributions on [0,1] generalizing the Beta family.

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