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We consider several special cases of iterations of random i.i.d. linearfunctions with beta distributed fixed points that generate nested intervalschemes when iterated in a backward direction, and ergodic Markov chains in theforward direction. We prove that the fixed points are beta distributed by usinga related random equation with a gamma distributed solution that also generatesa corresponding ergodic Markov chain with a gamma distributed stationarydistribution. Our approach allows us to find limiting distributions of therandom processes we consider, and provide solutions to the two randomequations, by solving only one of these equations. The paper extends manypartial results available in the existing literature.

On beta distributed limits of iterated linear random functions