Non-standard limit theorems for urn models and stochastic approximation procedures

The adaptive processes of growth modeled by a generalized urn scheme have proved to be an efficient tool for the analysis of complex phenomena in economics, biology and physical chemistry. They demonstrate non-ergodic limit behavior with multiple limit states. There are two major sources of complex feedbacks governing these processes: nonlinearity (even local, which is caused by nondifferentiability of the functions driving them) and multiplicity of limit states stipulated by the nonlinearity. The authors suggest an analytical approach for studying some of the patterns of complex limit behavior. The approach is based on conditional limit theorems. The corresponding limits are, in general, not infinitely divisible. The authors show that convergence rates could be different for different limit states. The rates depend upon the smoothness (in neighborhoods of the limit states) of the functions governing the processes.