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

Abstract

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: non-linearity (even local which is caused by non-differentiability of the functions driving them) and multiplicity of limit states stipulated by the non-linearity. 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. They show that convergence rates could vary for different limit states. The rates depend upon the smoothness (in neighborhoods of the limit states) of the functions governing the processes. Since the mathematical machinery allows us to treat a quite general class of recursive stochastic discrete-time processes, we also derive corresponding limit theorems for stochastic approximation procedures. The theorems yield new insight into the limit behavior of stochastic approximation procedures in the case of non-differentiable regression functions with multiple roots.