A Strong Law for Some Generalized Urn Processes

Key Points

• To establish the asymptotic convergence behavior and limiting distributions of ball proportions in generalized stochastic urn processes governed by continuous feedback functions.
• Modeled a discrete-time urn process where a red ball is added at step n with probability f(X_n), determined by the current red ball proportion X_n.
• Evaluated almost sure convergence of proportion sequences and characterized the support of the limiting random variable on the fixed-point set where f(p) = p.
• Demonstrated that the proportion sequence X_n converges almost surely to a random variable X with support strictly contained in the fixed-point set C = {p: f(p) = p}.
• Established that for any fixed point r in C under strictly bounded probabilities, P[X = r] is strictly positive when f'(r) < 1 and equals zero when f'(r) > 1.