A self-equilibrium Friedman-like urn via stochastic approximation

In this study, we propose a class of generalized Friedman urns with random entries. The class has the property of self-equilibrium. We prove the almost-sure convergence to the equilibrium point of this entire class by the method of stochastic approximation. We develop a central limit theorem for the proportion of white balls through the convergence theorem of stochastic approximation algorithms. We then apply this class of schemes to model the effect of trading on the price in the stock market under the efficient market hypothesis.