We consider a class of probabilistic models obtained by iterating random functions of $k$ random variables. We prove an analogue of the weak law of large numbers and under a symmetry condition we prove a strong law. The symmetry condition is satisfied if the initial random variables are exchangeable. Our results can be used to give stronger results than those previously obtained in the special case where the function is deterministic. Both types of models have applications in physics and in computer science.
"Almost sure convergence for iterated functions of independent random variables." Ann. Appl. Probab. 12 (3) 985 - 1000, August 2002. https://doi.org/10.1214/aoap/1031863178